What is the point of subspaces?

It is a very rare situation in mathematics to find oneself looking at a particular set and thinking, "hmm, I wonder whether this is a subspace".

In fact, this just about only happens when you're in a beginning linear algebra course and doing exercises. And the point of those exercises is not to train the skill of telling subspaces from non-subspaces (which in itself is pretty useless), but to give you an opportunity to develop an intuition for what a subspace is, and in particular what you can depend on if someone else gives you a set and promises, "oh, by the way, I've made sure this is a subspace".

A lot of subsequent concepts and results in linear algebra are formulated in terms of subspaces, so in order to properly understand and internalize those results one needs to be absolutely familiar with the idea of what a subspace is -- not just on the level of being able to reproduce the definition on command, but on the level of having an immediate idea what can and cannot be expected of a subspace.

For example, in order to speak about dimension you need a subspace. If you have a system of linear equations, the solution set is a translated subspace. Eigenspaces are subspaces. Spans are subspaces. Orthogonal complements are subspaces. If you have a linear transformation between vector spaces, the image of a subspace is a subspace, and the preimage of a subspace is also a subspace -- in particular, the image of the entire domain is a subspace, and the preimage of $\{0\}$ is a subspace. This alone can teach you a lot about the structure of sets in a vector space where concrete calculations may be difficult to carry out or even imagine intuitively.

The set of all solutions to a linear homogeneous differential equation: $$a_n(x)\frac{d^ny}{dx^n}+ a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}}+ \dots+ a_1(x)\frac{dy}{dx}+ a_0(x)y= 0$$ form an $n$ dimensional subspace of the space of all $n$ times differentiable functions. That means that means that, while there are an infinite number of functions satisfying that equation, if we can find just $n$ independent solutions, $y_1(x),$ $y_2(x),$ $\dots,$ $y_{n-1}(x),$ $y_n(x),$ then we can write any solution, $y(x),$ as $$y(x)= C_1y_x(x)+ C_2y_2(x)+ \dots+ C_1y_1(x).$$ The crucial point is that the definition of "subspace" the sum of two such functions or such a function times a number is still in the subspace.

The set of all solution to the linear non-homogeneous differential equation, $$a_n(x)\frac{d^ny}{dx^n}+ a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}}+ \dots+ a_1(x)\frac{dy}{dx}+ a_0(x)y= f(x)$$ do not form a subspace so we cannot form new solutions by adding old ones or multiplying by numbers.

It matters for the same reason why it matters that something is a vector space: you have the entire theory of linear algebra at your disposal.