What causes air to move to combustor?
What forces air to move from lower to higher pressure?

In the compressor, the pressure continuously increases from first stage to last stage, that's the purpose of the whole thing. But your question is valid: Why would air move from the first stage to the second stage where the pressure is higher?
The reason is indeed the rotating blades of a stage push air to the next stage. Energy extracted by the turbines is used to rotate the compressor blades and create a relative wind. Air is accelerated over the blade and like for a wing, a downwash force is created. This accelerated air is then slowed down by the next stator vanes to convert the additional velocity into additional pressure, and maintain velocity nearly constant in the compressor (else efficiency would decrease), as visible in tsg's answer.
Stop the blades and air goes back until pressure equalizes in all stages.

I have heard seals are used to prevent leakage of air from compressor
to forward section of engine

Correct, there are seals. But while the air moves towards the combustor within the compressor annular duct, under the effect of blades rotation, seals are here to prevent it from getting back to previous stages by escaping the annular duct, and leaking below the stator vanes (between the rotor drums and the inner shrouds of the vanes).
The fixed vanes plane is actually the plane where pressure increase occurs. So air will try to return on the front side of the vanes, towards the rotor blades using the interstices (orange flow), but it will be blocked by seals.
The seals are highlighted in yellow on the picture below, they rub on the inner shroud of the stator vanes (actually for the engine shown, a honeycomb ring fixed inside the vanes shroud provides an abradable track to the labyrinth seal:


HP compressor interstage labyrinth seals (CFM56-7B), source
These seals are required because a small gap exists between the rotor and the stator platforms forming the inner wall of the duct.