How do mutations with very little fitness benefit get fixed?
In the long run (over hundreds or even millions of generations), small statistical advantages are bound to accumulate.

Even if a mutation gives only a +1% average reproductive advantage, it is still above chance and therefore this mutation is more likely to spread through the population.
(If you can code, I have found it to be an interesting exercise to write scripts that simulate generations of evolution.)

It is also not necessarily true that effects of mutations have to be small. Single mutations can be deadly, they can change colors, create large morphological changes, and so on. It just happens to be the case that most mutations that have large effects are deleterious, and therefore won't generally spread for long throughout the population. But some mutations will be beneficial and therefore have improved odds of being passed on.