The rim elevation is the finished elevation of the top of a structure, such as a manhole or catch basin, where it meets grade. It is given in feet above a sea-level datum on the civil plans, not as a height off the ground.
For a precast producer, the rim elevation and the invert elevations together set the overall height of the structure and how the sections stack. Get the rim wrong and the top of the structure sits high or low against the road. It is one of the numbers a shop drawing has to nail exactly.