[zornthewise]

The simple idea is that while linear maps(matrices) are generally a combination of rotations, scaling and other stuff in higher dimensions, sometimes they will act on a vector simply by scaling it.

Since scaling is a really easy operation to understand, the space generated by these vectors will be really easy to understand. Note that different vectors can be scaled by different amounts.

Now it often turns out that the space generated like this is actually the whole of the space under consideration and this really simplifies the linear map we started with. Hope that helps.