This answer will refer only to reversal films. B&W does respond similarly, but at a much lower level.

Let's take a simple case, E6 film exposed to magenta light. Since the film is made of layers sensitive to yellow, magenta and cyan light, we expect that the magenta layer only contains exposed silver halide crystals (the latent image). During normal development only those exposed silver grains will develop to form a magenta dye cloud around each grain; the cyan and yellow layers will have no developed dye, and will be transparent.

Now we will underexpose, then push process our film. Push processing means to leave the film longer in the first developer (which acts much like a B&W developer), about 15% longer than normal. Since the film is in the first dev a longer time, silver grains that would normally not respond to the developer during normal development now have additional time, and some of them will respond. This means that some silver grains in the yellow and cyan layers will develop and form dye clouds around the sensitive but unexposed grains. In the fully-developed film, instead of seeing pure magenta, there will also be a bit of yellow and cyan 'polluting' the pure color. Saturation is the dominance of one color (be it a single-layer color like M, C, or Y, or a dual-layer color like red, green or blue), and any diminishing of that dominance, by mixing in other colors, is a loss of saturation.

But if it's only a single-stop push, the effect isn't very noticable.

The increase in grain during push development comes (I think) from one exposed silver grain influencing unexposed grains nearby, so that they develop in a cluster instead of individually. This will form big dye clouds that we see as grain.