Roman (and Ancient Greek) grammarians seem to have thought of verb paradigms somewhat like noun paradigms: the forms of puella "girl" are puella, puellae, etc, and the forms of amō "love" are amō, amās, etc. Rather than listing out all the forms, you can refer to the whole paradigm by its first element: the nominative singular for nouns, the first person singular for verbs. This convention is used by Varro, among others. There's not anything special about the first singular as opposed to other persons and numbers; it just happens to be the first on the list. (And the choice to put "first person" first and "second person" second and so on seems to go back to Dionysius Thrax.)
English, and most other Indo-European languages, tend to use the infinitive as the citation form ("citation form" being the technical term for the way you list it in the dictionary). But this isn't universal. For a couple examples of modern languages, Bulgarian uses the first singular present; Arabic uses the third singular past. From personal experience, I can tell you that Hittite uses either the first singular present (ḫarkmi) or third singular present (ḫarkzi), depending on the source; Akkadian uses either the infinitive (parāsum) or the third singular masculine past (iprus); Egyptian uses the "root" rather than any specific form (sḏm).
In the end, it comes down to what a particular lexicographer thinks will be most useful. For Latin, this is almost always the first singular present, because of the weight of tradition; that's the form Latinists expect to see. For something like Hittite, where there aren't hundreds of years of tradition for how to organize dictionaries, an individual lexicographer has some freedom to make their own choices.