This is a fascinating topic, and it really boils down to the mechanics of lift. Physically, upwash and downwash are generated as the air accelerates/decelerates and curves around the LE of the wing. At the wingtips, higher pressure air curves upward and generate the vortices we are familiar with. In fact, there are vortices generated at the trailing edge of the wing, too, just not as noticeably. Obviously, this picture is unsatisfying since it doesn't give us any insight to lift.
According to the potential flow theory (irrotational, incompressible and inviscid), lift is generated whenever there is a net circulation of air. A particularly successful model, the Lifting Line Theory, models the span of the wing as a collection of U-shaped horseshoe vortices: there's a bound vortex at the span location, and two trailing vortex arms extending to the far field.
So there you have it. A tiny downwash is generated by an individual vortex as it turns the flow downwards at that location, and a tiny upwash is generated as it turns the flow upwards. If you proceed to do the math and sum over all the influences at any point on the span, the net effect is a downwash (which changes the effective AOA and gives rise to induced drag as you mentioned).
If you do the same for a location ahead of the LE, you will get a net upwash, and inverse for TE. This seems intuitive: if you zoom out and treat the whole wing as one single circulation, the circulation should be clock-wise (assume air comes from the left), with air moving upward to the left and moving downward to the right.
There are more complex models, like vortex lattice, vortex panels, etc. But this theory is fairly intuitive and introduces the induced drag. By the way, induced drag is a finite span phenomenon. If you have an infinite span, you fully recover the 2D lift coefficient and the induced drag disappears, even though the vortices remain.