Langlands' functoriality conjecture predicts that to a suitable homomorphism of $L$-groups $$ \psi : {}^LG \to {}^LH $$ there should be a transfer of automorphic representations from $G$ to $H$. For the purposes of discussion, let's take $^LG$ to be the Weil form $$ ^LG = \hat{G}(\mathbb C) \rtimes W_{\mathbb Q} $$ where $W_{\mathbb Q}$ is the Weil group of $\mathbb Q$. This conjecture, as we know, has revealed many connections between disparate objects in representation theory, geometry, and number theory, and also works to explain various phenomena that we observe. My question is more on a philosophical level: setting aside the reasoning along the lines of "we believe it because it works," why should functoriality be true?
To narrow the question a little, what is the meaning of the $L$-group? How should we think of the semidirect product? What category does it live in? It blends a complex reductive group with the arithmetic of $\mathbb Q$, which is crucial to the entire framework of the Langlands program. As Casselman pointed out here, Langlands' letter to Weil already established that Langlands understood the centrality of the $L$-group, but this fact seems to have revealed itself through Langlands' deep experimental knowledge of Eisenstein series. Later work in geometric and $p$-adic Langlands reveal that the geometry of the $L$-group certainly realizes functoriality in certain senses, but I don't think it quite explains (for me, at least) the question of why.
The picture gets even muddier if we replace $W_\mathbb Q$ by the conjectural automorphic Langlands group $L_\mathbb Q$ as Langlands' reciprocity conjecture (perhaps) demands.
EDIT: To clarify a little further based on David Loeffler's answer. I realize that on some level it is a little bit of a fool's errand to ask such a meta question, but I will try to justify it. Certainly after over 50 years after Langlands' conjectures there is little doubt that they should be true, and as was pointed out, the $L$-group seems to arise in some natural sense especially in light of the Satake isomorphism (and its geometric variant too). This is along the line of what I mean by "we know it because it works." I think what I am trying to ask is in what sense might Langlands' Functoriality principle (as Arthur calls it) be more like an actual functor than just a principle? Here I am thinking of the usual local/global Langlands correspondences (which Langlands calls "reciprocity") as the special case of functoriality where $G$ is trivial. So at the base level, we have a functor from $$ \{\text{admissible $L$-homomorphisms of $L$-groups}\} \to \{\text{packets of automorphic representations of reductive groups}\} $$ up to necessary equivalences, in a way that captures reciprocity as a special case, as Langlands originally formulated. (I understand that $p$-adic Langlands, among others, has discovered much more intricate data and Arthur's conjectures too, so I'd be happy to receive input on how to update this picture. Inded, people working on questions related to modularity have thought a lot about category theoretic, and nowadays derived, approaches, but not at the level of Functoriality as far as I kow.)
But the basic question is to what extent can we understand this in a more category theoretic way, so that this map might be an actual functor? With this in mind, this seems to lead quickly to the question of how should I think about the $L$-group arises from trying to make sense of the left-hand side in some meaningful way. Of course people have sought to study things like the stack of Langlands parameters, or quasicoherent sheaves on $\text{Rep}(^LG)$, but this all still seems to take the $L$-group for granted (with good reason of course), but if I think of the LHS as a homs of a category, what kind of category am I looking at? Is there some topological or geometric way in which it arises "naturally"?
