Why care about group actions?

Solution 1:

I believe it's more the other way around: a group action of $G$ on a space $X$ allows to construct a new space, the quotient $G\backslash X$ (of course the quotient space is "nice" only under some technical conditions).

Recognizing that a space $Y$ is actually realized as the orbit space $G\backslash X$ of some simpler space $X$ can help understand better the geometry and other features of $Y$.

For instance, realizing the real projective plane as the quotient of the sphere $S^2$ by the antipodal action of the group with two elements, makes evident its non-orientability and gives immediately a concrete realization of the non-trivial element in the fundamental group.

The simplest cases are possibly those of the circle $S^1=\Bbb R/\Bbb Z$ and the torus $T=\Bbb R^2/\Bbb Z^2$.

There is actually an important situation where the action of a group $G$ is helpful to gain information on $G$ (not $X$), namely when $X$ is a linear space and $G$ acts via linear transformations. This is the object of Representation Theory.

Solution 2:

$\def\R{\mathbb{R}} \def\SL{\text{SL}} \def\SO{\text{SO}}$Often you use the group action to study $G$ and not just to study $X$.

Here is an example: what does $\SL_2(\R)$ look like as a manifold? You can solve this by thinking of the group directly, but an easier way is to note that it acts transitively on the upper half plane by Mobius transformations. Since the stabilizer of $i$ is the circle group $\SO_2(\R)$, we get $$ \SL_2(\R)\sim \mathcal{H} \times S^1 \sim \R^2 \times S^1. $$ (not an isomorphism of Lie groups!)