I don't think it is too much an overstatement to say that nobody has any idea why the most general conceivable form of the modularity conjectures-say a combination of Langlands program and the Fontaine-Mazur conjecture-should be true. As in the case of conjectures on special values of $L$-function, the most one could probably say is that their inner consistency is absolutely impressive so that in some sense, they feel too good to be not true.
That said, not all is lost, I think, in your quest to get a philosophical understanding of this topic, especially if you set yourself a more modest goal at first. Because why things should be true is probably inherently subjective, I will only offer my personal experience with modularity results for $\operatorname{GL}_2$. I think that the first significative experience I had towards a modicum of understanding of the deep reasons why these should be true was to realize how utterly surprising they were. The more I understood about abstract universal deformation rings and the less I could see why they should be Hecke algebras. The Taylor-Wiles method, I still don't claim any deep or philosophical understanding of, but this is mostly because I never read closely enough the literature. Some papers from Kisin, for instance, do explain that there seems to be a trade-off between how singular a deformation ring can be and the local behaviour of the Galois representation at p. The next big step for me was to read carefully Taylor's paper on potential modularity. This paper makes it very clear that modularity results are very amenable to bootstrapping: prove one, and you may get a lot for free. So to recap: modularity results should be true because (in certain settings), one can reduce them to much simpler modularity results and then get rid of the singularities of the universal deformation ring (provided you have what you need to do so).
Not very philosophical perhaps but since the name of Weil has appeared in the answers by Emerton and Joël, let me conclude by quoting his magnificent (if slightly depressing) words on finding philosophical understanding of mathematical theories.
Rien n’est plus fécond, tous les mathématiciens le savent, que ces obscures analogies, ces troubles reflets d’une théorie à une autre, ces furtives caresses, ces brouilleries inexplicables ; rien aussi ne donne plus de plaisir au chercheur. Un jour vient où l’illusion se dissipe ; le pressentiment se change en certitude ; les théories jumelles révèlent leur source commune avant de disparaître ; comme l’enseigne la Gita, on atteint à la connaissance et à l’indifférence en même temps. La métaphysique est devenue mathématique, prête à former la matière d’un traité dont la beauté froide ne saurait plus nous émouvoir.