I think that even the most ardent supporter of minimalism should recognize that this is an important and deep question: indeed, even though Patrick Elliott is right to recall that the hypothesis of strict binary branching could be made on the basis of parsimony only, it is clearly the case that many syntactic structures appear to be ternary (or more) and so that the hypothesis that all syntactic structures are built from the bottom-up by recursive applications of a binary operation is extremely counter-intuitive and should never be assumed lightly. In fact, this hypothesis is quite recent in the literature of generative or transformational syntax: as far as I know, it was first argued for by R.Kayne in 1981 and gained dominance even within the relevant strand of linguistics only in 1994/1995, so after more than 30 years of hierarchical analysis of syntactic constructions.
When trying to organize this answer, I realized that I would face two obvious difficulties: 1) binary branching is a theoretical hypothesis, with immediate theoretical consequences but with quite sophisticated and elusive direct empirical consequences and 2) the analysis of empirical facts is not independent of the implicit or explicit theoretical framework as a wide range of contradictory interpretations can describe equally correctly any given single phenomenon (as can be seen for instance in Tim Osborne's criticism of bta's answer). A geocentrist will not be impressed by the fact that Newton's law of gravitation explains why the orbits of planets around the Sun obey Kepler's law as for him there are no orbits around the Sun to explain; and conversely, a heliocentrist will not be particularly convinced by the remark that adding a seventh epicycle yields unparalleled experimental accuracy to the prediction of the apparent movement of Jupiter. Hence I decided to present empirical evidence in the form of correlations, that is parallel occurrences of syntactic phenomena, whenever I could with the accompanying claim that the theoretical arguments I am evaluating are able to predict this correlation or not: the empirical evidence below therefore do not reside in any specific analysis of, say, reflexivity in English or of wh-movement in Japanese, but in the fact that an analysis of reflexivity in English allows to make correct predictions about wh-movement Japanese. That said, one has to start somewhere, so I will assume without further discussion that tree-like descriptions are useful in syntax: anyone subscribing to a completely linear theory of syntax will presumably not care about whether (for him non-existent) syntactic trees are binary or not, after all.
I think the first step to understand the empirical arguments in favor of binary branching is to understand the peculiar importance the notion of c-command has come to gain in (a certain kind of) syntactic theory. A number of seemingly disparate syntactic phenomena (licensing of pronouns and anaphors, quantifiers scope, NPI licensing, wh-movements, clitic movements and reduplication, licensing of cataphoric ellipsis…) all turned out (under some interpretations) to be sensitive to that specific topological relation. Yet why would that be?
For instance, independently of any theoretical presuppositions, one can observe that both
(1) It seems that Tatsuki is angry because Masako bought something but I don’t know what.
(2) The picture of himself that Tatsuki took is nice.
are grammatical in English whereas their direct Japanese translations
(3) *Tatsuki ga Masako ga nanika o katta kara okotteiru rasii ga, watashi ha nani ka shiranai.
(4) *Tatsuki ga totta zibun no shashin ha kirei. (with the interpretation that zibun refers to Tatsuki)
are not. The apparent causes of ungrammaticality seems very different for (3) and (4) yet a linguist who believes that anaphor licensing and wh-movements are both sensitive to c-command is able to give a unified account for this (at least in the sense that he would predict that (1),(3) are grammatical if and only if (2),(4) are). Likewise, from the acceptability of
(5) Nos vamos acostumbrando a este país poco a poco.
(6) Vamos acostumbrándonos a este país poco a poco.
in Spanish, he might have been able to predict the respective acceptability of
(7) Vámonos acostumbrándonos a este país poco a poco.
(8) *Nos vamos acostumbrándonos a este país poco a poco.
in Argentinean Spanish, or at least would have been able to formulate the prediction that the acceptability of (8) would logically entail the acceptability of (7). Or again, his hypothesis that wh-movement would have to be successive cyclic from c-commanding positions to c-commanding positions would not by itself predict that Afrikaans realizes overtly copies of the wh-word as in
(9) Met wie het jy nou weer gesê met wie het Sarie gedog met wie gaan Jan trou.
Whom did you say again that Sarie thought Jan is going to marry?
but given that they are, they turn out to be precisely where the c-commanding positions have to be in comparable sentences in English or French based on any of the other diagnosis above.
Of course, a number of alternate explanations and descriptions can be given for any of these facts taken in isolation: the specificity of the c-command account is that a single principle motivates the causal relations between each of them.
But why should c-command, by opposition to any other topological property starting with the much simpler rule of sisterhood, play such a significant role? Here we have two concurring but quite distinct theoretical answers. Kayne’s answer is that absent stringent supplementary formation rules, very few hierarchical structures can be converted unambiguously into linear sentences. Binary branching structures with asymmetric c-command satisfy this desirable property. So in fact, according to Kayne at least (but the idea seems largely accepted in minimalist syntax) not only is binary branching the unique mode of branching allowed, asymmetric binary branching structures are the unique structures allowed, yielding the prediction that no binary structures like
(10) [A B]
are actually possible even though they are an obvious model for coordination, for instance, and the even more counter-intuitive prediction that the number of functional projections in a given syntactic trees should be roughly half the total number of nodes (as follows from some math that I skip here); seemingly predicting the existence of far more functional projections that would be assumed based on a superficial analysis of the sentence. Chomsky’s answer is that the simplest linguistic operation we can imagine to construct hierarchical structure is binary (asymmetric) set-theoretic union of labelled terms (with the asymmetry coming from the projection of one of the the labels). The recursive application of this operation predicts the importance of the c-command operation, in a way that is worth summarizing as it will play a role later on: under this model of bottom-up recursive applications of a binary operation with no supplementary operation, interactions between heads are by construction limited to interactions between a head and its complement and the easiest type of movement to conceptualize is movement from the complement to the specifier of the head. This gives movement to a c-commanding position as the simplest syntactic operation available and more generally strongly suggests that elementary syntactic operation should occur between c-commanding positions.
Empirical support for binary branching beyond c-command
So binary branching is not primarily empirically motivated, it is a theoretical simple assumption which has the desirable effect to predict the special status of c-command in syntax, yet there would be no point in formulating theoretical hypotheses if they did not suggest empirical insights. Here are a few. First, the binary branching hypothesis suggests that the importance of c-command is not a syntactic fundamental but rather the reflex of the easiest possible relation between heads. Hence, one predicts, for instance, that as long as the functional heads assumed to be involved in anaphor binding are in mutual c-command relations, binding ensues even when antecedent and anaphor are not in a c-command relation. For what its worth, just like the prediction that wh-movement was successive cyclic, this was a genuine prediction in the sense that it was articulated and argued for before any example of the phenomenon was known. It turned out to be correct.
(11) Det ble introdusert en mann for segselv. (Norwegian)
It was introduced a man to himself.
The precise mode of relations between heads that is implied by binary branching also predicts the relative acceptability of
(12) What did you order? What did Sophia order?
(13) What did no one order?
(14) Toi, tu as commandé quoi ? Sophie, elle a commandé quoi ?
(15) *Personne, il a commandé quoi ?
even though the c-command relations are all exactly the same.
As a further example, the extremely strong constraints on structures imposed by strict application of asymmetric binary branching implies the hypothesis that much more hierarchy than is usually visible or even plausible at first glance should exist. To give an example, the hierarchical structure Kayne was forced to posit for nominalization such as
(16) The election of Obama.
implies that the complement of of c-commands the leftmost phrase at some point. Again, this was a genuine prediction, in the sense that the empirical data confirming this prediction were later found (in Dutch).
(17) Het over zichzelf praten van Jan.
The about himself talking of Jan.
(18) Het aan zijn eigenaar teruggeven van elk geleend artikel.
The to his owner returning of each borrowed article.
Of course, countless further phenomena have by now been analyzed in this way from morphology to person-case constraints to parasitic gaps to wh-agreement.
Binary branching is a theoretical hypothesis, and as such should not be expected to have direct empirical consequences, much less direct empirical consequences which can be recognized as such from a theoretical framework different from or directly contradictory to the one it presupposes. Within its own theoretical framework, it predicts the ubiquity of the c-command relation while predicting where the c-command relation will yield incorrect predictions, with a broad range of empirical consequences; anaphor binding outside c-command, weak-cross over effects in nominalization, intervention effects in left-dislocation and island effects in elliptical questions being just a few sampled here.