"Efficient" means "the difference between benefits and costs is maximized". "Transactions costs" are the costs of transacting--the costs of bargaining to reach agreements. They could be language barriers, regulatory differences, excessive arrogance or mistaken beliefs of the parties to the agreement, or even coordination failure due to a large number of participants on one or both sides of the negotiation. "Rights" could mean any right to anything--the right to graze one's sheep in a particular area, the right to be free from pollution, the right to pollute, or even the right to host a television show at a particular time.
The recent mess over at NBC over late night television is, I think, a simple example. NBC and Conan O'Brien signed an agreement years ago, in which NBC promised that Conan O'Brien would host the Tonight Show. It is not clear whether the contract guaranteed O'Brien the 11:35 time slot, but let us assume for the sake of argument that it did (even if it doesn't, The Tonight Show is closely associated with that time slot, so that O'Brien might be able to argue that it isn't the same show if it airs at a different time).
Eventually O'Brien took over The Tonight Show from Jay Leno, who got a new show at 10:00. Leno's new show performed poorly. O'Brien's Tonight Show also did not perform as well as NBC had hoped (in particular, NBC's affiliates were especially upset).
Now NBC has a problem. They have promised O'Brien The Tonight Show and (perhaps) its time slot. Leno also wants his old show back, as his new show is about to be canceled. What efficient new bargain can they strike?
It is pretty straightforward, and it is the result that the network and O'Brien are rumored to have struck. NBC buys O'Brien's right to The Tonight show, and lets Leno return to the show. Everyone is better off: The viewers and the NBC affiliates get the show they want (bizarrely--I can't imagine preferring Leno to O'Brien). NBC should expect an increase in profits as a result. Leno gets his old show back, and O'Brien gets a large amount of money, and the opportunity to go to a network that will appreciate him. (If you must have some numbers, suppose the network values having Leno back on The Tonight Show at $60 million, and suppose O'Brien values staying on the show at $20 million. There is a range of offers that would make both parties better off.)
Similarly, if NBC had never promised the show to O'Brien, this same outcome would have occurred--Leno would have the show, and O'Brien would not. The parties end up at the same efficient outcome, regardless of the starting allocation of rights.
What about transactions costs? There could be transactions costs that prevent a bargain from occurring. O'Brien might be offended and stubborn, or he might overestimate how much NBC would be willing to pay for the rights to host The Tonight Show. If he asks too much, NBC declines to pay. Similarly, NBC might underestimate how much O'Brien loves the show, and might try to offer too little. Still, both these problems should be addressable with repeated negotiation. The more serious impediments to bargaining don't seem to exist here.
A side note: I am leaving out wealth effects; I think this is reasonable in this case, given the wealth of the parties involved.
Another side note: I suppose this is also an example of efficient breach.
UPDATE: John Lentz points out that it appears transactions costs were low enough for an efficient bargain to be struck.