In my opinion (because it really is an opinion), the main reason to care about euclidean spaces is that they come equipped with a lot of particularly nice structures. In particular, they are finite-dimensional Hilbert spaces. This means that they are:
- Finite-dimensional vector spaces (so we can talk about addition and scaling)
- Complete metric spaces (so we can talk about distances and limits, and limits behave as we would like them to)
- Inner product spaces (so we can talk about the notion of "angle," and thereby do all sorts of geometric things.)
And really, there's just so much geometric intuition that comes along with these ideas -- not to mention an entire calculus apparatus. After all, there are plenty of topological spaces where the above notions are not defined or otherwise fail to be true.
The reason (at least to me) to study locally euclidean spaces is that we want to study spaces that are more general than euclidean spaces, yet still retain many of their nice features. In particular, we want a place where calculus makes sense.
Areas of math which study more abstract spaces include topology and algebraic geometry. Admittedly, I'm not very well-versed in either just yet, but I'm sure practical uses (and physical models) have been found in both.