Squid expand the mantle cavity by contracting sets of muscles within the mantle, water fills the expanded space, the muscles relax, and the elastic mantle then snaps back to a smaller size, jetting water out through the funnel. The jet of water closes the flaps on either side of the squid's head so water can exit only through the funnel. This rhythmic flow of water is also the way in which squid breathe. As water passes in and out, the gills are refreshed with oxygen. But let's back up.
Muscles get shorter when they contract.
Squid mantles are thick. Muscles within the mantle actually compress the thickness without collapsing the outer skin. The mantle then behaves like a syringe, and water is drawn or, more accurately, pushed in. The second way squid move is with great bursts of speed and acceleration, which they do by recruiting other muscles to make dashes through the water. Essentially, they get more water into the mantle and they push it out more forcefully.