Trypsin in principle is considered to be a protease. What a normal protease does is to "cut" the animo-acid bonds by hydrolysis. One vivid example of a similar enzyme is lysozyme; it is located in human saliva and helps us with our food process by hydrolysing the starch that exists in food.
So, you need trypsin as an protease to get them detached from the cell culture wall of the dish. This way the cells will be "swimming around" in your medium and you will be able to transfer them to your new cell culture dish with fresh nutrients.
In general, due to its protease function, trypsin shouldn’t be left for more than 30 sec in contact with your cell culture otherwise your cells will be teared apart and I guess you don’t want this to happen.