And what does either the change of scenery or the lack of competition have to do with the random mutations of DNA? This is the part that always got me. DNA is not a sentient being in organisms, its not like it chooses what to do to itself. Sure, there are certain proteins that change the DNA strand, if there is a mistake on it, they don't make the mistakes. Weather I need an extra arm or a striped back pattern is irrelevant to the DNA strand. I can't consciously or subconsciously control my DNA strands. It is genetic. Any mutations of the DNA strand are usually harmful or have no effect to the organism.
Who said anything about changing the rate of mutation -- though that too can be changed easily enough. Whenever there is a major catastrophe resulting in multiple extinctions, there is both a large environmental change and also several empty niches.
Normally, the theory of evolution predicts that beneficial mutations are quite rare -- after all, the organism has been evolving for millions of years and all the easy good mutations would have already happened and been selected for. But after a major catastrophe, the organism is no longer adapted to the environment, and therefore the rate of beneficial mutations must increase. Suddenly the organism may be better off with a change in body size, diet, etc and since there are many new options available many of the mutations which would previously have been bad are now good mutations.
Over and over, whenever there is a catastrophe new species quickly evolve to take the place of those extinct. (Note that quickly still means millions of years.)
Then at what level does it occur?
Oh, evolution occurs at all levels -- groups, individuals, gamets, the cells in your body, etc. Always populations of such, of course. The traditional, simplified version is that evolution "only" occurs in populations at the level of individuals, and that is also where most of it does happen.
However you can have evolutionary competition in your own gamets, for example, where it is called "meiotic drive". These cases are very extreme, and can have mutations that cause a 50% drop in fertility and yet have a 90% chance to be passed on.