The only way to enable a cutscene is via commands using command blocks, command prompts, etc. You can say:
"Timeline = (Timeline ID)"
You'll likely need to fine-tune your timeline if you haven't tested it out yet.
To enable it at the start of the game's title screen, simply go to Title Screen settings, and enter the afore mentioned command into the "Area Start" command prompt. That's going from memory, so the command prompt might say something else. To start at the begining of each world, that depends if it's a world map or a hub world. Commands aren't supported for the hub world because none of them are compatable with it. You could create a hubworld, or if you really want to, you can re-create the world map in the level editor.
The way to work around the problem that the world map doesn't support commands would be that you can set it up so that once you press the start game button, it'll take you to the area with the cutscene, and then once it's finished, to the world map. Or whatever you have setup. You can use variables to cause it to take you to a different world so that when a world is complete, it'll store a variable number that PB will check and then in turn make decisions on the number result. If you don't know where variables are, you can go to the tutorials section and read TingThing's article on them. (It's a sticky topic, so you won't have trouble locating it.)
Anyway, welcome to the forums, Fi! We hope you make some great games in the future, and thanks for choosing Platform Builder! Don't forget to write an introduction in the introduction's board if you wish.
Happy Platform Building!