What is the process by which organisms with desirable traits are more likely to survive?

I'm not sure what your question was, but I'm thinking you meant "WHY DO organisms with favourable traits have a better chance of surviving?". That's what I've answered below.

Creatures which are better adapted for their immediate, local environment have a better chance of surviving because they are more able to overcome adverse situations specific to that environment. For example, if a creature eats a specific type of food and that food is only found at the top of tall bushes, a favourable trait would be being able to reach that food. Having these traits not only means that they have a better chance themselves of surviving, but it means that they have a better chance of living long enough to reproduce, thus passing on their genes. A prospective mate may recognise favourable traits and choose one mate over another so that her offspring may inherit those favourable traits and will have a better chance of survival.