Air pressure pushes down on the hydraulic fluid, the fluid is not compressable so the pressure energy is transfered to the hydraulic fluid. This allows you to do without a hydraulic pump and the other mechanisms needed to pressurize hyd fluid. This is a very simple and cost effective way of producing a "soft" acting hydraulic system.
Not very scientific but I hope that helps, really it is so simple. What specific aplication are you unsure about?
Here's what I don't get. If I take a piece of vertical pipe filled with oil, plumb it to a hydraulic jack and an air line I will end up with a rudimentary air over hydraulic system. I will also end up with wet hydraulic oil.
I understand the theory -- air PSI x surface area of hydraulic piston = force but what about the mechanics of the system? I've never actually been into one.
Along the lines of what Jim said, consider a large diameter air cylinder. Lots of area, some air pressure and you have a bunch of force on the rod.
Hook that rod to the rod of an opposed small bore hydraulic cylinder. Your large load from the air cylinder can produce a larger hydraulic pressure.
Give the hydraulic cylinder a simple check valve so when it strokes at high pressure, the fluid that's pumped out is stuck behind a check valve. When it goes back home it fills again from the reservoir, which can be done from another check valve that's opposed in direction to the first one.
The air cylinder can be made to "pump" or cycle continuously with a little valve that trips at the appropriate time to shift the pressure and exhaust ports on the air cylinder.