I remember being constantly reminded of this by all my teachers as a child. I just thought of it as an annoying little thing I had to do to get to playing what I wanted to play. Thankfully he was very insistent and I had no choice but to pay attention to it constantly in the beginning.
Unfortunately very few teachers bother with any explanations to justify such a universal rule of thumb (sorry ). And I myself only began to see it's value when I had to come back to the basics of bow hold much later. With self analysis and slow practice on difficult passages I began to notice that tension always starts accumulating when I forget to pay attention to my thumb muscle.
The thumb is attached to the palm with the biggest muscle in the hand. Other fingers are primarily controlled by tendons that stretch up to the big muscles in the forearm. Tension is built in the muscles and a straight thumb is a sure way to lock up and build up such tension; especially while playing fast and difficult passages.
Since the natural (default) position of all fingers is slightly curved, this neat little trick goes a long way to release tension before it becomes a problem. As our first instinct when encountering real difficulty in instrumental playing is to tense up, it is especially useful to pay attention to the thumbs; both in the right as well as the left hand.