I live on a dairy farm, so I'm going to use that for examples.
Let's start with genetics. When I'm getting ready to breed a cow, I want to use a bull that has a reputation for producing high-quality, high-producing animals. That reputation suggests that he carries good genes, and I want the calves of my cow to have those genes as well. I also don't want to inbreed. Inbreeding can bring out negative recessive traits that could compromise the health, longevitiy, or productivity of my calves. Additionally, a severely inbred animal could be born with any manner of physical deformities, as intense inbreeding will weaken the integrity of the genetic strand.
I could also use genetic engineering (this is more of a hypothetical; genetic engineering is used more in meat livestock, and even then not terribly frequently). If someone isolated a gene that increased milk production in some way, I might decide to have that gene implanted in a calf embryo.
There are also options in the dairy industry to sex embryos. Female animals are more valuable in the dairy industry for the fairly obvious reason that males don't produce milk. So by determining the genetic sex of the sperm I'm using or the embryos I'm implanting, I can ensure that my herd will grow more quickly.
OK, so now I've picked the bull I want to use based on his genetic history. Unfortunately, he's on a farm 1000 miles away. Fortunately, reproductive science has provided us with artificial insemination, and I can simply order a test-tube of our bull's, ahem, essence, rather than transport an animal across the country.
Of course, it's very important to me, as a farmer, that my now pregnant cow produce a healthy calf, and that she stays healthy throughout the reproductive process. Not only is it just sad if a cow or calf dies during birth, but it is not good from the business end, either. Luckily, veterinarians can check the progress of the pregnancy to ensure everything is happening normally, Furthermore, during and after birth, we can intervene should anything happen that may put the cow or calf at risk (say, a breached birth). Basically, reproductive science makes fertilization easier, and birth safer and healthier.
Now that my calf is born, nutritional science has taught me that his mom needs extra calcium to keep her strong right after the stressful labor process. I also know that a calf that is fed it's mother's milk for the first few days of his life will develop fewer diseases. After that, I know what the best options are to feed him in his various life stages. Since he's a bull calf, he will be fed to produce that tastiest beef. His sisters would be fed to keep them strong during lactation. Proper nutrition is important not just to keep their production high-quality, though, but to ensure they'll live long lives and stay healthy. The better nutrition an animal gets, just like a person, the faster they'll grow, the stronger they'll be, and the more able they'll be able to fight off diseases.
These can all be generalized to pork, poultry, and whatever other animal you can think of. The general idea behind all of these things is to produce as many of the strongest, healthiest, and most productive animals as possible.
Nutrition provides the better understanding of food and fodder for animals and their protection from diseases.
Reproduction helps to increase the number of off springs.
Genetics allows us to obtain desirable characters in live stocks.
Hence, all above said branches are useful for the better livestock production.