I understand two reasons why reductive groups play such a central role nowadays.

First:

A maybe not so satisfying answer to "Why are reductive groups important in the Langlands program" is: $\operatorname{GL}_n$ is important.

It happens that many conjectures and motivating ideas in the Langlands philosophy for $\operatorname{GL}_n$ can be *easily* formulated in the more general setting of arbitrary reductive groups. Proving things in this more general setting is another matter entirely.

I don't believe there is any definition of reductive group which motivated Langlands to formulate his conjectures that way. Rather, the structure theory of arbitrary reductive groups is so similar to that of $\operatorname{GL}_n$ that the important ideas for the general linear group carry right over.

An example is parabolic induction. A reductive group $G$ has various Levi subgroups $M$, which are reductive groups in their own right, but generally less complicated than $G$. Many representations of $G$ are parabolically induced from representations of Levi subgroups. So the idea is that if we understand representations of "smaller" reductive groups $M$, we will know more about representations of the bigger group $G$. This is especially evident in the case $G = \operatorname{GL}_n$, where the Levi subgroups look like products of smaller $\operatorname{GL}$s.

Second:

Number theoretic objects associated to arbitrary reductive groups are expected to be special cases of those associated to the general linear group. I sketch this argument below. Breaking up these objects associated to $\operatorname{GL}_n$ into subcases might be insightful.

If $^LG$ is the Langlands dual group of a reductive group $G$ over a field $k$, $r: \, ^LG \rightarrow \operatorname{GL}_n(\mathbb C)$ is a representation, and $\pi$ is a representation of $G$, there should be an associated L-function $L(s,\pi,r)$. The Langlands correspondence should (roughly) associate to this a homomorphism $\rho: \operatorname{Gal}(k_s/k) \rightarrow \space ^LG$. This correspondence should preserve the L-functions

$$L(s,\pi,r) = L(s, r \circ \rho)$$

The composition $r \circ \rho$ is an $n$ dimensional Galois representation which by the Langlands correspondence for $\operatorname{GL}_n$ ought to associate this to a representation $\Pi$ of $\operatorname{GL}_n(k)$. That Langlands correspondence should also preserve L-functions:

$$L(s, r \circ \rho) = L(s,\Pi)$$

and therefore

$$L(s,\pi,r) = L(s,\Pi)$$

So the philosophy is that everything should eventually come back to $\operatorname{GL}_n$.