How can dna mutations and genetic variation lead to the changing of a population over generations (evolution)?

Well, heres an example 

If birds eat the beetles that have green wings, and suddenly a gene for brown wings is produced the beetle population as a whole will breed for brown wings so their population can continue to grow.

If we didn't have this mutation or genetic variation the beetle population would eventually die

(The example is made up)  

Of the cold they can survive in the cold

Yes yes yes very good