With my right hand, I leave it on the backing paper and slowly drag it to the right (pulling it away from the mylar).  Now, I always get the missus to help and she will either stand at the end of the playfield holding the two end corners of the mylar up, ensuring I am keeping everything square. - This is important if full playfield.  Sometimes she may also drag the backing paper from the other side to keep it square as well.  With my left hand, with the terry cloth I'm rubbing it back and forth - full width of mylar just where it touches the playfield.  Drag with one hand, rub with the other....

I hope this makes sense?

I've got quite a few off cuts I can send you if you want.  If you have a piece of MDF or ply laying around you can do a couple of test runs.  Once you've got the technique worked out is pretty easy as long as you go at a very slow pace and keep the mylar sheet square and even.