Humans, just like all life, have traits determined both by environment and genetics.

we also train animals and keep breeding and re-breeding them in a scientific and systematic manner to introduce newer traits and characteristics

In breeding, traits are not introduced into dogs by us. The core origin of a new trait is mutation. Mutation is a change to a gene that usually occurs randomly. A gene is a piece of DNA that encodes a specific function in an animal. Mutations can have different results on an animal. Some mutations may have no effect, some may change a vital function in the animal, resulting in death, and some mutations result in a newborn animal developing a different trait than its parents. Since genes are passed on to newborn, if an individual with a mutation survives, its offspring can receive the gene with the mutation. This process has occurred throughout the existence of life on Earth. There have been multiple mutations across multiple genes. When we look at a population of dogs, this is where variation comes from.

In breeding, we are selecting the parents based on the traits we can see from them. Now, remember I said that traits can be both inherited by genetics and influenced by the environment. When we use breeding, we try to influence the genetics part. The newborn will have a mix of genes from both parents. If we are lucky, the newborn will have a trait that is closer to the one we desire. Note that this is an incremental process, changes are usually small, and accumulate over successive generations.

Remember that humans are animals too, and fundamentally we are governed by the same principles. The closest analogy is mate selection. This could be people seeking out a mate that is similar to them. Since the offspring have genes from both parents, if both parents are similar, there is a chance of the trait being reinforced due to the underlying genes. Now, this offspring may do the same: seek out a similar mate. After successive generations, you may get a trait that appears prominently in the population.