Almost all mutations have little or no effect. You have a dozen or so that weren't in either of your parents. A few are harmful, and tend to be removed from the population. And a very few are useful, and tend to spread through the population.
It's directly observed to work that way. It was a serious problem for evolutionary theory in Darwin's time. Scientists thought at the time, that heredity was in the blood, and it seemed that a new trait would be swamped out of existence in a population like a drop of red paint in a barrel of white paint. Only after Mendel's work was discovered, did they realize that heredity is like sorting beads, rather than like mixing paint, and the problem was solved.
In general, evolution never produces something entirely new; it always modifies something already there. Often, one structure, such as a walking leg, is reconfigured over time to a new function, such as flight, digging, or swimming. This seems to never have happened suddenly; there are always transitional forms showing a gradual change with the limb serving more than one function at least for a period of time. Would you like to discuss that further?
For example claws in amniotes (reptiles, birds, and mammals) are entirely different than "claws" found in other vertebrates like the Sea Robin (fish) and the clawed frog. Each of these is genetically and histologically different than claws of amniotes.
The earliest likely amniote, Casineria, had rudimentary claws. Since claws are modified keratinous structures that are histologically and genetically like scales, it's likely that this tetrapod had body scales which were modified at the digits to form crude grasping structures. Existing structure, reconfigured to a new use.
Why do some harmless mutations survive? Because there's no selective force to remove them. However, over a long period of time, such mutations tend to either become fixed (all individuals have them) or disappear. If you do the math, you'll find that for a finite population, random events will very likely result in extinction or fixation over a long period of time.
I've got some work in entomology, and ants are an interesting case. Where there are more than one species in an environment, they are either adapted to non-competitive ways of life, or one is replacing the other. The arrival of fire ants in the American South was a disaster for many native ant species, but over time, they seem to have adapted to the intruder, and some are coming back. Others may be doomed.
The issues are much more complex than you suppose that they are. The field in which this is studied, is called "population genetics." Would you like to learn about it?