To "be slammed" at work means to have a lot of work to do. The word "slam" is also used to describe the sound of a door being shut quickly, or of someone falling and hitting the floor. So you can imagine "being slammed" as having a large amount of work suddenly hit you while making a large sound.

"Slammed" is an ugradable adjective, like "horrible", "exhausted", "delicious", or "enormous". These are all words that have extreme meanings, so it seems strange to say that you were "very slammed" or "a little slammed".