The multimeter tests continuity by sending a little current through one probe, and checking whether the other probe receives it.
If the probes are connected—either by a continuous circuit, or by touching each other directly—the test current flows through. The screen displays a value of zero (or near zero), and the multimeter beeps. Continuity!
If the test current isn't detected, it means there's no continuity. The screen will display 1 or OL (open loop).