If someone has “A lot on their plate“, it means that they have of stuff they need to do. They are very busy. Someone could say, “I’d love to go for a holiday somewhere with you but I’ll have to wait until the new year because I have too much on my plate right now at work”.
Someone could say to their boss, “I’d love to help lead the new project, but I have so much other stuff on my plate right now that I really need to finish first.”
A husband could call his wife and tell her on the phone, “Honey, I’m sorry but I’m going to be late again tonight. I need to work overtime. My boss has put so much on my plate lately and there is some urgent stuff I need to finish by tomorrow”
So having “a lot on your plate” means you are very busy and have a big “to do” list.
Imagine that you have a plate filled with so much food. There is no room for new food on your plate until you eat some of the food that is currently on your plate. That’s the idea of the background meaning of this phrase.