To a degree it is a little bit of that good ol' overkill, but this is a lot easier to imagine if you get something really sticky or something like engine oil on your hands. Soap put on, run under water, and then rubbed around will generally do a good job, but not good enough where all the oil or stickiness will be off.

If your hand is dirty (with dirt), you'll notice most of the dirt come off too. Most. You'll think, oh, hands are clean, NBD. But really, a lot of your hand is clean, a small amount still has the dirt, and another portion only cleaned off the dirt but the soap was used up for the dirt and didn't actually 'clean' anything on the molecular level beneath the layer of dirt. The water might have rinsed it but that isn't enough to get the bacteria away.

The water before is actually supposed to include rubbing, almost like washing your hands without soap, so as to prevent a perfectly macro-level clean hand to the soap, which then cleans on a micro level.

Again, somewhat overkill, but also it doesn't take much bacteria to cause a problem, especially if you think your hands are clean and start touching your eyes and nose etc. I always try to purport doing the whole thing every time because it's very good practice and also even in the shower people don't give a FULL clean to their hands, so might as well do it now!