I speak in generalisations here, because there's huge scope for tuning it with varying weighs, reps and ballistic lifting techniques etc, but in simple terms...
Squats increases the size of the blood vessels. It promises the growth of new ones.
This increases the capacity to pump blood to that muscle.
During exercise that muscles demand for oxygen increases, and the increased blood supply meets that need more effectively, granting that muscle the ability to work harder/ longer before the supply depleted.
The increased blood supply also removes wastes more efficiently, and thus slows the rate at which lactic acid builds up, which is itself the cause if the unpleasant burning sensation one feels when pushing a muscle hard enough for long enough. The longer you can offset that reaching your own individual pain threshold, the longer you can make that muscle work. This is the big one as its the changing perception of effort as the burn increases that limits most people's endurance, not the muscles mechanical ability to continue.
That's very simplified, but fundamentally is it really.