Geometric interpretation of this bilinear interpolation equation?
If we look at the two terms with only x or only y they fully describe what happens when you go right or up from the starting point.

The xy-terms does not come into play at all.

When you want to go "into the square" or along the upper or the right edge then the xy-term comes into play. It cancels out the x/y-terms and introduce the P11-value that it should obtain in upper-right.

No full explanation here, just nudging along towards insight!