for the take off at a high density altitude, the air is thinner so you have less mass being moved around by the wing and therefore less lift at a given (true) airspeed. likewise your propeller is producing less "lift in the forward direction" which means less thrust.
Also, your engine is getting less oxygen and so your engine produces less power to turn the prop.
Indicated airspeed is measured by the difference between dynamic pressure and static pressure. the density is on both sides of that equation and so it cancels out. therefore, you get the same amount of lift for a particular indicated airspeed regardless of density, but it takes a lot more speed to make that lift at a high density altitude.
humidity affects density, but it is a pretty small factor. the thing to worry about is heat.
you don't even need to be full fuel and heavy to have problems due to density altitude. once in an older 172 with only two people on board and half tanks I had to abort takeoff on the big runway at PRC, 5045 feet and 90 degrees.