OK, let me explain it from ground zero.
There are many different meanings of the word "rent" in economics, but the #1 modern usage is "An earning in excess of opportunity cost." A worker earning $10 an hour, when their alternative on the open market is merely $9, is considered to earn a $1/hr rent. (Why use the word "rent"? Well, it all goes back to Ricardo and other classical economists. Since the land is just "there," they figured that from some point of view, the opportunity cost of land is zero. The term rent then got expanded to apply to anything "land-like"; i.e., any resource that is "just there" and the existence of which does not depend upon its being paid.)
The general assumption is that rents are just useless inefficiencies. They are basically just like the government granting a monopoly on salt; the price of salt then exceeds its opportunity cost, and for no good reason.
A quasi-rent is different.
It LOOKS a lot like a normal rent; from a superficial viewpoint, it is a reward paid to a factor which exceeds its opportunity cost. But if you look deeper, you see that in fact, the rent is a necessary incentive for something. For example, a patent looks a lot like a monopoly salt grant on the surface. The pricing policies of the patent-holder and the salt-monopolist look a lot alike. But the crucial difference is that the patent-holder had to do something to GET the rent; he had to develop a new product. In the long-run, no industry can earn more than the normal rate of return, so the effect of the patent is just to get more people to try to invent products; enough people so that inventing things earns the same average rate of return as anything else. What do quasi-rents compensate for? The obvious example is innovation.
A lot of innovations are hard to patent, but an industry leader like Microsoft can be reimbursed for its R&D by winning a leading position in the market. Other good candidates include search (a worker earns more money because looking around for better jobs takes effort which must be compensated).
Also -- quasi-rents may be paid for product variety. When products are heterogeneous, there is room to charge a little above opportunity cost; but the incentive effect of this "breathing space" is to encourage the satisfaction of consumers' diverse tastes.
Of course, it would be more efficient to pay a one-time, lump-sum "bounty," and then price the product at opportunity cost; but that usually isn't workable. The real world system is pretty damn good, once you give it a chance to understand it on its own terms.