It is like when there is a huge crowd moving in one direction, and if you get into the crowd you will be pulled to go with the crowd. We say that you 'get swept up' because you are 'carried away' or inspired to move along with something, or to feel sympathetic for it. It describes a feeling as being infectious, like a circus attracts children, maybe pulling them away from something else they were doing. We also say this to describe when we become so involved with something that we forget about the time. We say 'I got swept up' in the subject, or we 'got caught up in' the subject that distracts us.

To add to what Zooey explained... To sweep is to use a broom to scrape the floor and push all the dirt to one side. "Swept" is past tense of "to sweep". The dirt was swept up by the broom, and you were swept by the excitement so you went to dance like all the other people. When you say you were "swept up" by something, it means that you let yourself go, to experience the moment, and feel the full force of it. You can be swept up in the movie and cry for the main character. Or you can be swept up by the strong emotions of an election, or sporting event, or birth of a child, etc.