The toe clips and straps hold your feet on the pedals, so whatever energy you use to keep your feet on the pedals will be saved; that increases efficiency and reduces fatigue.

They don't fit onto every pedal, though. They're a great solution with the right shoes, something with not a lot of height. MTB shoes, for instance, are hard to use in clips.

I've used toe clips and straps, toe clips, straps, and cleats, SPDs, and pinned flat pedals. I've never raced. The best compromise I've come across is pinned flats and Five Ten shoes, because that combo gives me the most surface area, which minimizes pressure per unit of area on my foot.

I never fell with clips and straps (many bike seasons), fell once with clips, straps, and cleats (one season), and fell innumerable times with SPDs (1/4 of a season). I never fell with the pinned flats, but once or twice a season they punch holes into one of my calves.