The competitive advantage of having eyespots is small, perhaps even negligible, on the individual level. It might decrease chance of being eaten by 0.5% (made that number up as an example). It is when you have millions of insects in an area that the advantage becomes apparent: 0.5% of 1 million is 5000. So every generation of moths with eyespots is expanding at a slightly faster rate than their cousins without eyespots, until eventually the cousin species might be driven to extinction.

(Disclaimer: I'm tired, the math and description above is a bit shady but works as an illustrative example. There's more to it than that)

I think it helps to understand that whilst it is mostly the genes which confer a competitive advantage surviving through the generations, not all genetic mutations do confer competitive advantages. A mutation is a random change, and the new genetic information could do a number of random things. So with eyespots, a few random mutations created dark spots on the moths wings, and over the course of thousands of years it happened to be these moths which were surviving in greater numbers because animals are stupid, and a lot of them ran away from the 'eyes' on the moths' backs. Then more mutations occurred, and it just so happened that the random patterning looks a bit more like an eye.

Wrote this on a phone, sorry if I'm rambling.