As jsw29 notes in a comment beneath Ryan M's answer, the source idiom is "have too much [or a lot] on one's plate." Here is the entry for that expression in Christine Ammer, The American Heritage Dictionary of Idioms, second edition (2013):
have a lot on one's plate Also, have too much on one's plate. Have a great deal (or too much) to cope with, as in What with the new baby and the new house, they have a lot on their plate, or I can't take that on now; I have too much on my plate already. This expression transfers a loaded or overloaded dinner plate to other activities. {First half of 1900s}
Often the expression arises in connection with a request for help from someone else, as in "I have too much on my plate right now. Can you help me out by taking my place at the Bowser-wowser demo this afternoon?" But in some cases, as the posted example suggests, ridding oneself of excess responsibilities may entail simply dropping them rather than transferring them.
Another form of the source idiom is "have a full plate." Here is the entry for that expression in Cambridge Dictionary of American Idioms (2003):
a full plate a lot of work to do or problems to deal with | Facing funding cuts and a lawsuit challenging the school's admissions policy, the university's new president has a full plate.
The implication of the metaphor is that if you already have a full plate, you can't add anything to it without first getting rid of something that is already on it.