Say you know the coordinates of all the corners of a cube in three dimensions. If you wanted to move that cube to the side and turn it, you could first add some fixed number to all the x coordinates of every point, then do some trig to find the coordinates again after a rotation. Turns out you can produce the same output with fewer steps using matrix math. That is, you can produce a single matrix which is representative of the transform "move right m units and turn d degrees". And you can apply that transform matrix to any set of points you want.

At least, this is what I've gleaned so far as someone that doesn't really understand. I welcome corrections.

Another example is Vector projection [1], this is how video games (or design software) map 3D worlds onto the 2D screen [2].

[1] http://en.wikipedia.org/wiki/Vector_projection [2] http://en.wikipedia.org/wiki/3D_projection

You need quaternions to avoid gimbal lock. In my small experience with 3D graphics, you would only deal with linear algebra on building cameras.

http://en.wikipedia.org/wiki/Quaternion