How do Bowden cables work from a force transmission point of view? More specifically, the cable sheath plays a major role since the mechanism doesn’t work without it. So the sheath must be applying force somewhere, but I can’t understand how it really works.
Any help?
They are mostly used for pulling, as stated in the Wiki, so you might think of the hollow sleeve as a VERY closely spaced set of pulleys that the cable runs over. These pulleys automatically align themselves as required by directional changes in the running of the cable assembly. That’s very simplified, but it works for me. 
The cable’s sheath provides the actual mechanical linkage. The point is that the ends of the sheath maintain a fixed separation with respect to the length of the cable. So displacing the cable relative to one end of the sheath results in an equal displacement at the other end, regardless of the cable’s path in-between.
GaryM’s comparison to a pulley system is apt. For a pulley system to transmit motion accurately, the distance between any two pulleys must be rigidly maintained. But the pulley system is rather difficult to articulate (make bendable). The Bowden cable does the same thing with a bit more friction, but with much greater flexibility.
Two good examples are bicycle brake cables and camera shutter-release cables. The brake cable sheath transmits the compressive tension against which the brake cable pulls. The shutter cable sheath transmits the expansive tension against which the shutter cable pushes.