A while ago, dog breeders noticed that when they crossed breeding lines, the resulting mongrels would have more vigour, show generally improved health, and often live longer. This was duly noted, but since the focus was producing strains of dogs with specific traits (and there's no market for mutts), it was generally ignored. Starting in the 1960s and 70s, the role of DNA in evolution really began to come to the forefront of research, before it proceeded to explode in the 80s (with the advent of PCR and other molecular techniques).
Through mutation experiments, where they'd bombard fruit fly eggs with x-rays, people began to understand that the majority of mutations they could detect where `loss of function` mutations. Meaning, the x-rays changed a few bits of DNA, and suddenly the gene didn't make the stuff it was supposed to, or regulate the body function it was supposed to. The mutations they could detect in those days were generally bad.1
Now, since you have two copies of almost each bit of DNA (if you're an animal) - one from mom, one from dad - carrying around some `broken,` non-functional genes really doesn't hurt you. Lack dad's copy of the gene to make something? So long as mom's copy donated to you is intact, you're in luck, and you can still make it. In biologist lingo, the mutation is `recessive.` However, if both mom and dad's copy of the gene is damaged, you don't have any good copies of the gene. You can probably guess where I'm going with this.
Back to the pure-bred dogs. Humans have been controlling and breeding them for many generations, and they tend to breed with only related dogs. This has led to an elevated chance for these recessive, loss-of-function mutations to pile up in the bloodlines, and that's exactly what has happened. The smaller the breeding population is, the more likely they carry these bad traits. Which leads us to our mutts. When two dogs of different breeds mate, it's pretty unlikely that they both have the same non-functional bits of DNA. So your lovable mutt tends to inherit a working copy of just about everything from each parent - and this forms the basis for hybrid vigour and vitality in dogs.
What about wild critters? How does this work in the wild? Well, instead of humans guiding the breeding, it's nature doing the selection of who breeds with who. Similar to dog kennels, non-domestic animals tend to live in groups that are more likely to breed with each-other. Think of it like this: My neighbour is currently unlikely to marry anyone from California, for the simple reason that we're both in Alaska, and isolated by distance. The more isolated animal groups are, or the fewer individuals in their populations, the more likely that they carry some non-functional garbage.
For example. There are panthers in Florida. Strangely enough, they're called Florida Panthers. They're highly endangered, and there's not many of them left in world. Because of that, most of them aren't terribly distantly related to every other Florida Panther. Because of this, these deleterious genes started increasing in prevalence in their population, until it started making many of them less healthy. To counteract this, biologists in Florida took a number of cougars from a Texan subspecies and (highly controversially) released them into the Florida Panther population. The resulting Florida*Texan hybrids allegedly had more vigour and were healthier than either of their parents, in that environment2.
And that, deer readers (and human readers too), is the nickel tour of how Hybrid Vigour happens.
1 We've since revised our understanding, and now know the majority of mutations are neutral, and undetectable in fitness. Fewer, though still many, are harming mutations. A handful of mutations are beneficial to the organism.
2 The Florida Panther genetic rescue is highly controversial among conservation biologists, and discussion of it is a good way to start a fist fight at conferences and parties. Not everyone thinks that breeding the hybrids was for the better.