There seems to be at least two quite distinct questions there: what is minimalist about the minimalist program? and Is there a sensible reason for the existence of the lumbering structures it deals with? P.Elliott and curiousdannii already provided general and historic answers to the first question, but I wish to supplement their answers with a more specific analysis of the framework of a typical argument within minimalism as it is usually practiced.
What is minimal?
What a typical minimal analysis will strive for is an explanation of syntactic phenomena relying solely on the following list of possible mechanisms.
Bare syntactic structures constructed by the binary operation Merge (yielding only binary trees) usually with only left-adjunction by opposition to much flatter structure.
The only other operation allowed is Agree, which operates blindly on a small finite set of features all having only positive or negative polarity by opposition to e.g a rich lexicon.
Agree and Merge are only possible under stringent locality conditions.
The geometric and feature characteristic of nodes of the trees should be mapped as cross-linguistically severely as possible with interpretive properties.
As far as possible, the only further conditions put on the system should derive from core computational requirements (for instance two undistinguishable nodes of the tree, in the sense of say graph theory, or two identical set of features should be undistinguishable by the system; I also lump in this interface conditions).
This list can be deemed minimal because it is a negative list: it restricts the kind of explanation you are allowed to put forth. Especially, if one takes seriously point 4. and 5., it follows that the analysis of any construction in any given language (say left-dislocation in Spoken French) has to proceed through universal explanations in terms of geometry of the tree and feature properties, explanations which in turn possibly (and do, if the work is to have any value) imply predictions bearing on another totally different constructions (say wh-questions in Spoken French) or a similar construction in a totally different language (say left-dislocation in Japanese). For examples of what I consider good work done in this way, I would cite this or this.
Where do the lumbering structures come from and why they are actually empirical success stories for minimalism?
Now moving on to the justification of the lumbering structures. As Kayne first noted (as far as I know), the combination of point 1. and point 5. above implies the existence of many extremely refined functional projections (if only because a binary tree has a lot of internal nodes compared to the number of its leaves). So it is not that these functional projections were introduced, they were predicted to exist because they were essentially the only solution compatible with the imposed restrictions. This is indeed extremely reminiscent of the epicycles of Ptolemy and extremely worrying: if your theoretical framework leads you to postulate many things nobody has seen, shouldn't you be concerned? That's a very fair criticism but one which actually highlights the predictive power of the core principles above: if our prediction is correct, the functional projection posited (again as the only possible solution within the framework) have to be phonetically overtly present at precisely the assumed position in at least one language. The fact that this has been repeatedly shown to be true is one of the main scientific achievements of minimalism: any serious alternative account should face the challenge head-on and achieve similar predictive power within its own system. A prime example of such a successful prediction is successive cyclic movement (the point of departure between what became minimalism and many other formalization of generative grammar) which is phonetically realized (among may others) in Afrikaans and in cyclic agreement in Chamorro but one can also think about focal projection (realized in Vata) or voice projection (arguably realized in Japanese and Kiswahili). There are of course many more sophisticated ones, from binding theory, the structure of DPs, the properties of nominalizations, the logical interpretation of indefinite objects, the extraction properties of relative clauses... It thus seems to me that the comparison with angels on pinheads is quite unfair: these angels have been repeatedly found ex post on precisely the pinheads they were supposed to be dancing on.
What is the appeal?
To me, as a complete outsider to the field who has never taken a class in linguistics nor ever intend to (so I feel quite immune to any political influence this or that strand of linguistics might have on the curriculum or, a fortiori, on the hiring process), the appeal to this approach mostly stems from its cross-linguistic potential: as mentioned above, by nature, explanations valid in one language will make predictions and suggest insights for others. While it is quite easy to give a formal description of any specific linguistic phenomenon for a given language (any texas sharpshooter can do this), it seems to me that when it comes to the study of cross-linguistic correlations, minimalism is currently the only game in town. That said, I do think that the comparison with the Ptolemaic system is not off the mark: minimalism might currently be the most precise and clearest unified account we have of syntactic phenomena (just as Ptolemy's system, as famously argued by Otto Neugebauer, endured because of the clarity of its tenets and its very reasonable empirical basis) but it seems fair to me to say that linguistics is still awaiting its Kepler, not to speak of its Newton.