When to remove it from the stir plate and when to refrigerate it are two different questions. You can take it off the stir plate once active fermentation starts, if you want. Finishing out fermentation sitting on your counter won't hurt it. I sometimes take it off the stir plate after a day or two, once things kick off. Sometimes I leave it on longer. It's easier to tell when the yeast has finished fermentation if it's off the stir plate because the yeast starts to fall into a cake at the bottom. That's the point at which I refrigerate it, usually for a day or two, and then decant the spent wort and pitch.
You also don't have to refrigerate it. Some people pitch the starter during its active fermentation because the yeast is in the right mode. I don't like to pitch that much starter wort into my beer, plus I make larger starters and save half the yeast for future batches. So I always fridge and decant. But if you don't have that much time, you can always just pitch the whole thing. Once it starts forming a krausen, you should have the cell count you were aiming for.
