How exactly do species evolve over time?

The only thing I don't get is how a species develops different traits.. like say you have a species of wolves. and I get that by natural selection, the wolves will adapt to their environment and their DNA will change to code for traits that will benefit the wolves in a changing environment.. but how do the nitrogenous bases of the DNA strands just rearrange themselves to produce the amino acids for the proteins that will make up the traits that will benefit the wolves.. how do they just know what new trait to code for? ..and this can't happen during the wolves' lifetime (this was Charles Lyell's theory but it was proved wrong). My bio teacher said it occurs during ual reproduction, so it occurs with the swapping over of traits during meiosis? I don't understand this, can anyone explain it?