The key is to choose the right frame of reference, and all will become clear. Consider the operator that takes points (x, y, z) in 3-dimensional space and maps them to (x, y, 0). Geometrically, this is what user50618 suggested with the steamroller above: the x and y positions remain the same, while the z gets compressed down to zero.
The matrix representing this projection is:
1 0 0
0 1 0
0 0 0
If you put the column vector [ x y z ] to the right of that matrix, you see the multiplication:
1 0 0 x x
0 1 0 * y = y
0 0 0 z 0
If you do it again, no more change occurs. z remains zero. This is called being "idempotent."
And if you look at the diagonal of the matrix, you see two 1's for the directions that stay the same, and a single 0 for the direction that gets collapsed.
Now, ANY projection of 3-space onto a 2-dimensional subspace has EXACTLY that same geometry. It's just much harder to see when the direction of collapse is not lined up nicely with one of the coordinate axes.
The process of diagonalizing a matrix represents finding the natural directions of the linear transformation, and rewriting the matrix with respect to those natural directions. When written in a basis of its eigenvectors (the name for the natural directions), the matrix will be diagonal. In the case of a projection, the diagonal entries will all be ones or zeros, representing the directions which are preserved or collapsed.
Here is an example (not a projection), which is easy to write:
It is not immediately obvious what this linear transformation does, because its action is not aligned nicely with the coordinate axes. But think about what it does to the vector (1, 1). It collapses it to zero. And think about what it does to the vector (1, -1). That becomes (2, -2). This tells us that the matrix represents the same geometric action as this matrix:
That is, one axis gets stretched to double its length, and the other gets squashed down to zero. This becomes clear when you use the "right" basis - the basis of eigenvectors - to write the matrix.
Note that some matrices cannot be diagonalized, and some that can require complex numbers. But projections always can be diagonalized using only real numbers.