Why does more transistors = more processing power?
A lot of things that give you more power just require more transistors to build them. Wider buses scale the transistor count up in almost all processor components. High speed caches add transistors according to cache size. If you lengthen a pipeline you need to add stages and more complex control units. If you add execution units to help mitigate a bottleneck in the pipeline, each of those requires more transistors, and then the controls to keep the execution units allocated adds still more transistors.

The thing is, in an electronic circuit, everything happens in parallel. In the software world, the default is for things to be sequential, and software designers go to great pains to get parallelism built into the software so that it can take advantage of the parallel nature of hardware. Parallelism just means more stuff happening at the same time, so roughly equates to speed; the more things that can be done in parallel, the faster you can get things done. The only real parallelism is what you get when you have more transistors on the job.