It measures the “space” contained by an n-dimensional parallelepiped. In 2d, it’s called “area” and in 3d, it’s called volume. In general, it is just “n-dimensional volume”. Technically, I think the absolute value of the determinant gives the volume, and the signed value tells you about its orientation in space (ie, if it’s “below” an odd number of axes, it will be negative) but I haven’t really thought about that in a while.

I don’t know about the application of “determinant as volume” specifically. The determinant itself has many applications in various fields. Most obvious example that I can come up with is that it used in linear algebra to determine if a system of n equations can be solved.