Why study "curves" instead of 1-manifolds?

Essentially because the connected 1-manifolds are ("up to...") $(0,1)$ and $S^1$, so the notion of curve captures all of the possibilities of 1-manifolds sitting in higher-dimensional manifolds. In other words, the situation for 1-dimensional manifolds is so simple that it really makes no sense to use the full machinery of embeddings and immersions to talk about them other than just checking that the definitions of embedding and immersion are compatible with the definition of curve.

Also, in agreement to what kahen said:

In a regular parametrization, you can always find a metric that is constant along the curve leading to zero intrinsic curvature. This is also a reason why it might be somehow not very enlightening to discuss curves as one dimensional manifolds in its own right.

Curves, on the other hand, can have extrinsic curvature which can for some body moving along a trajectory be interpreted as the acting force.

Greets

Robert