The "master" cylinder is connected to your clutch pedal - when you press it - it moves the hydraulic fluid (brake fluid) in the master "cylinder" through the hose and into the "slave" cylinder.
By pressing fluid into the slave cylinder, it forces the slave to expand to accommodate this additional fluid. When it expands, it pushes the throwout bearing attached to it forward into the fingers of the clutch and frees the clutch plate to spin free of the flywheel and pressure plate.
So, the slave together with the master accomplish the same thing as a clutch cable does on an older model. The fluid and cylinders on either end of it (master and slave), act like a "cable". The benefit being automatic adjustment, better lubricated moving parts, and depending on the configuration higher mechanical advantage (less pedal pressure for the same or more throwout bearing pressure).