The .w component clicked for me when I understood that it allows to transform 3D-vectors (describing a direction and magnitude, .w is 0) and 3D-points (describing a position in space, .w is 1) by the same 4x4 matrix. The translation part of the matrix is multiplied by the w component. Thus a vector (with .w=0) will not be translated, only rotated and scaled, while a point (with .w=1) will be scaled, rotated and translated.

All the esoteric projection- and 4D-stuff is really not that important compared to the simple feature of allowing to feed positions and directions into the same matrix operations.