Although purpose-made motors and generators differ in detail, fundamentally they are identical machines that can serve each other's roles up to a point. What determines whether a machine works as one or the other is where the power is put in (mechanically or electrically) and taken out. Connect a machine to the grid and attach a brake, and it will run as a motor converting electrical power to mechanical. Replace the brake with a Diesel engine and the same machine will run as a generator.
A phase converter has multiple windings and does both at once. The winding energised by the single-phase supply receives power and functions as a motor, turning the rotor. The rotation causes the non-energised windings to function as a generator, from which current (and hence power) is delivered to the load. Part of the load power is converted from electrical to mechanical and back to electrical. The phase relationships of the three output phases are created by the mechanical positions of the windings relative to the rotating flux vector in the converter, just as in a mechanically-driven generator.
That is true as far as real power flow is concerned. There are many subtleties and complexities about phase converters due to the reactive power flows, but you will need some AC circuit theory knowledge to analyse those and the functions of the capacitors in a practical converter setup.