Remember how a reply gets written: one token at a time, each one picked straight after the last. With no scratch space, the model has to get a tricky answer right in a single pass, with no chance to try an approach and back out.
Thinking gives it that scratch space. When thinking is on, Claude works through the problem in thinking blocks first, then writes its answer. It can try an approach, check it and change course. That helps most with maths, logic and multi-step jobs.