Dried-out compost piles will eventually break down, but it could take some time. Keep your pile moist, but not too soggy. The dampness of a wrung-out sponge is ideal for organisms to turn plant materials into finished compost.
Turn your pile as you spray it with water to evenly coat all materials. If you find it too hard to turn, make deep holes in it, stick a garden hose down them and turn on the spigot. Leaves should glisten with moisture and shredded paper should be wet, but not mushy.
Hot, dry climates may need a little extra water to reach the right moisture level. Let nature help out. If your weather forecast calls for rain, leave the top off of your compost container to enjoy the shower. But be sure to replace the cover after the storm to retain heat and moisture, as well as to keep out pests.