ErikdaRed is right. When the studios park was planned, it was felt that Disney did not have enough content to open such a park. Disney therefore paid for use of the MGM name and access to the library for attractions, themeing, etc. I have read over and over that Disney received MGM usage rights for a paltry sum because they negotiated at a time when MGM was desperate for cash. Supposedly, the CEO of MGM was so enraged by the deal that he threatened litigation, etc. but Disney already had a legitimate contract.
There has been ongoing speculation that the deal with MGM is due to end soon and will not be renewed. IMO, who cares - Disney has enough content now and 'Disney Studios' sounds better.