To "load up" something means to put stuff on or in it until it gets full. Aside from your dishwasher, some other things that you can "load up" include:
load up your truck
load up your plate (with food)
load up a basket
There's a very small difference between just using the word "load" and using "load up". When you say that someone "loaded" something, it makes me think of the action of them putting things on or in something. But when you say that they "loaded up" something, it makes me think of the result of having a full dishwasher, truck, bag, etc. However, the difference is small so you don't really need to worry about which word to use in most situations.