This is due to the fact that the oxygen is used to burn the fuel placed in the chambers of the engine. When more oxygen can be placed in the chamber it allows one to also place more fuel along with it and therefore create more power. A 800cfm carburetor takes in 800 cubic feet per minute of air. If the air is denser than there is more of it in 800 cubic feet and therefore more amount of oxygen will be taken into the engine in one minute. Hopefully, you now understand what density is.
Density altitude is the density expressed if feet instead of grams per cubic centimeter.
The use of density altitude is taken from the U.S. standard atmosphere table. As one climbs in altitude the density falls off at a predetermined exponential rate. This is one way an aircraft can know how high it is. Recently, oxygen sensors have been being used with weather stations. Sometimes the percentage can vary a few tenths of a percent.