The verb congest means to clog up and become blocked. It is frequently applied to a head cold, when your nose begins to congest. During rush hour the roadways begin to be congested, as they become blocked and vehicles slow down.
We know that "con-" means "together," and the Latin congestus, which is the source of the word congest, means "collected, heaped up." Something that is congested can certainly be thought of as heaped full of stuff. By 1758, the word had taken on the medical meaning of an "accumulation" — think of your nose during a cold! — and the idea of congest as referring to something "overcrowded" was noted in 1859, giving rise to the idea of overcrowded cities as congested.