They raise and lower the motor vertically. Some have a greater set back which gives you more stability. Being able to raise it allows for safer running in shallow water. But I guess the most common reason for using them is more speed. You can get them fixed, electrically operated or hydraulic operated. If you get an adjustable it is highly recommended you install a water pressure gauge. Outboards don't go very far without water getting in them for cooling.
Fair Winds and Following Seas
Bill H. PTC USN Ret
Chesapeake, Va