You could have it where the cutscene is triggered by completing the previous level. The very last event could be something like "On 'something happening', go to 'cutscene layout'".
Another option would be to trigger it at the start of the new level and have both the cutscene and the new level on the same layout, but this might be a little more difficult to manage unless you're really organized.
For the actual cutscene it can be anything, like what TL22 said. You could have images that fade in and out, maybe have them move a little and text fade in/out. Or you could do animation if you have the patience to do it.