"Moved on" is two different words. It isn't the phrasal verb 'moved on'. The cars were moving. Where were they moving? They were moving on the street. The cars were not stopped on the street. Below grade could be a lower floor, but in general it means that the bottom of the door (and the floor it leads to) is below the road, or path level. You have to step down, or go down steps to get to it. I assume that 'set flush' here means that the surface of the door is level with the wall. in other words, the door is not set into the wall, or sticking out from the wall.

You have asked for the meaning of "moved on" but in this sentence the words are actually divided as follows: "the cars moved" + "on the street" = "cars were moving along the road" ---> the pairing of the words "(to) move on" means for you to forget about a sad/troublesome situation and stop being emotionally affected by it. "below-grade door" refers to a door below street level, in other words, a basement door that is accessed by stairs at street level going down. This is quite the technical terms, as far as I'm aware though. "(to be) set flush with..." means that the two objects being referred to (the door and concrete wall here) are perfectly aligned with each other.

A below-grade door means a door located in the basement of a home/house. Moved on means, the cars drove by. (The cars passed or were driving around)