How are new people created from the DNA of an aged person. i.e. Why are we young?
Gametes (sperm and ovum), which fuse to form a zygote, arise from germ cells (spermatogonia and oogonia). Germ cells, like stem cells, are maintained carefully i.e the genome is preserved and transposition/recombination events are tightly controlled via different mechanisms. 

So these germ cells don't have shortened telomeres. Also, during early embryonic development the future germ cells quickly migrate to an end of embryo and remain there while the rest of the body takes shape. 

I'm not sure if it has been demonstrated in an article or not but once I happened to talk to Dr Wakayama (who cloned a mouse frozen for 16 years) and got to know that animals cloned by the process of Somatic Cell Nuclear Transfer have reduced lifespan. (SCNT: the nucleus of a somatic cell is transferred to an ovum and triggered to form zygote). 

So, to conclude, if germ cells were not protected from mutations and deleterious recombinations, lifespan would have been reduced with every generation.