Why is my Macbook Air MagSafe Pin Becoming Extremely Hot?
This can happen when the MagSafe cable is failing, generally at the plug end. As the cable strands break down due to mechanical wear, the power being supplied has to be carried by fewer strands. These strands invariably begin to heat up and become oxidized, increasing resistance and therefore generate yet more heat. This is why your MagSafe is getting hot.

If you're handy with DIY, repairs to the MagSafe connector are possible (at least with the old white plastic MagSafe connectors). Another option is to crop off the old connector and splice on a replacement cable and connector - these can be found at the usual internet retailers. Finally, you could simply buy a replacement power supply.

Sadly, Apple show no sign of environmental responsibility with their designs and the MagSafe power supply is no exception. They can only be opened with brute force, invariably resulting in something that can no longer be regarded as "safe". You're lucky in that it looks like your cable has failed at the connector end, which can be replaced. It's just as common for Apple adapters to fail at the adapter end, which is almost always goodnight Vienna for the adapter.