Studies of traffic dynamics rely either on macroscopic models considering the traffic as a fluid, or on microscopic models of drivers' behavior. The connection between the microscopic and macroscopic scales is often done via empirical relationships such as the fundamental diagram for macroscopic models, relating traffic flow or average velocity and traffic density. In this paper, we consider a microscopic model consisting of a large number of rational, utility-maximizing drivers interacting on a single road. We then use the theory of Mean Field Games (MFG) to deduce a macroscopic model of traffic density emerging from these interactions. We show how to determine a microscopic utility function for the drivers compatible with standard empirical macroscopic fundamental diagrams. In addition to connecting the microscopic and macroscopic models analytically rather than empirically, our approach can offer additional flexibility to model drivers at the macroscopic level, using a Hamilton-Jacobi-Bellman equation coupled with the standard conservation law for the vehicles.