Great question. I am not one of the science experts here, but I'll give a brief response.
Genetic mutations happen all the time. At some point, perhaps, I'll tell you my personal story with unexpected genetic mutations. But let's just start with that.
The vast majority of mutations do not change anything substantial in a species. But every once in a while, a mutation changes a specific characteristic. Maybe it changes the color of an animal's fur, which results in the animal's ability to camouflage itself. The animal that is able to camouflage itself is more likely to hide from predators, and therefore is more likely to live to reproduce, and more likely to pass on that color of fur. That animal's offspring are more likely to have that color fur, and are more likely to survive, and more likely to pass the fur color on to their offspring. Within a few generations, nearly all of the surviving animals will have the new fur color.
So organisms do not evolve to survive. Organisms undergo mutations, and some mutations change certain characteristics. If a specific characteristic increases the survival rate of members of that species, chances are it will be passed on to future generations, and ultimately the species will change.
Does that make sense as an introduction to the concept? I am sure that there are lots of resources here that members can share with you.