It measures possessions used. Each possession in basketball is valuable, each one that player A uses is one that player B cannot. So there are some players who have high usage but low efficiency (like nick young) and some who have high efficiency but low usage (like Kyle Korver). Nick Young or whoever (just going off the top of my head) can average 15 points per game but use 18 possessions to do it. While Kyle Korver could average 12 points per game but only use 11 possessions per game. So in this therotical example, Korver leaves 7 possessions available to his teammates and all they have to do to match Young is get 3 points on 7 possessions, which is very easy to do, thus making Korver the more valuable offensive contribution despite scoring less points.
Usage rate is a percentage stat so if everyone was equally used each player would have a 20% usage rate. (There are 5 teammates on the floor and if they divide 100% of the possessions equally they each get 20% to use). Now obviously some players are more ball dominant and some are just spot up shooters or guys who score on offensive rebounds.
There are guys who help their team by being high usage (Steph Curry) because they are efficient and guys who hurt their teams by using so many possessions when they can't do it efficiently (Rusy Gay). Same with low efficiency guys, Anthony Morrow is an efficient player because he's a good spot up shooter so he helps his team when he takes corner threes. But he can't dribble drive or create his own shot so even though he is efficient, he can't really increase his usage rate without taking bad shots and lowering his efficiency. Then you've got guys like Tony Allen who are on the floor for their defense or Ricky Rubio for passing. These guys are bad offensive players and their efficiency is low but at least they are smart enough players to understand it and limit their possessions used (thus low usage, los efficiency).