The contacts in the outlet have lost the abillity to squeeze against the prongs inserted into them. This causes a poor connection, which is a high-resistance connection, and due to the high current passing through to the dryer, causes a lot of heat.
This should not be happening, unless the plug has been removed and installed many many times. The plug just sitting in the socket does not cause contact wear.
How much heat ? If the dryer uses 20 amperes, and the resistance of the outlet-plug connection is one-tenth of one ohm, then 40 watts of heat is generated - and it only takes one of the contacts inside the outlet to cause this problem. That's a lot of heat in such a small space !! If the resistance is one-twentieth of one ohm, the 20 watts of heat is generated - still a lot of heat. Think about how hot incandescent bulbs like that get.
It is possible some of the wiring to the outlet has been damaged by the heat and that section of wiring will have to be replaced.
The wiring is not hard to do. Take photographs of everything as it is removed, and then install the replacement components in the reverse order. You can find do-it-yourself electrical books at hardware stores, libraries, and used-book stores that show how it is done, generically.
You can get a new outlet and the needed replacement wire at any large hardware store, just take the burned outlet (that matches the new dryer plug) to the hardware store to get a matching replacement. The store should cut-to-length any wire you need to replace any overheated wire.
EDIT :
I just throught of another possibility. If the house is between 30 to 40 years old it might have aluminum wiring in it. If there is aluminum wiring between the breaker panel and the outlet, that aluminum wiring will have to be removed and replaced with copper wiring. There were a huge number of problems with aluminum wiring getting loose under the screws or clamps inside of sockets, which caused overheating and sometimes fires. Aluminum wire is more ductile than copper wire and under the pressure of the screws or clamps it would "creep" as it expanded and contracted with temperature changes. Thus the thickness of the aluminum wire would get smaller, which caused a loose connection to begin. Furthermore, aluminum forms a skin of aluminum oxide on its surface where oxygen gets to it, and this skin is an insulator. This forces current to flow through a smaller area of the aluminum surface that is in contact with the copper or brass connectors of the outlet, which means higher resistance, and more heat. The copper oxide that forms on the surface of copper wires is conductive and not as big a problem.
Aluminum wiring was a huge mistake.
http://en.wikipedia.org/wiki/Aluminum_wire
Aluminum wire