It's pretty clear how signing a message with a Bitcoin key can be useful in proving ownership of funds.
However, it's far from clear how encrypting a message with a Bitcoin key can do anything useful. This feature is available from Tools/Encrypt/decrypt message, and appears to use ECIES or some variant of it:
It seems that at least one developer is doubtful about the security of such schemes:
Security aside, what practical reason would anyone have to encrypt a message with a Bitcoin key?