When kittens are young, they are raised by their mothers, who often bring back food for their kittens.
This feeding behavior is deeply imprinted in the kitten's mind. When the kitten grows up and meets someone it wants to take care of, it will imitate the mother cat's feeding behavior and give them the prey it has caught. 
Sometimes owners get startled by their cats. They might be quietly doing something when their cats suddenly rush over and scratch them. Sometimes they'll jump really high, staring intently at sparrows in the distance. More often, they'll stay up late at night to catch mice. These are all instinctive behaviors of their hunters.
Deep love for its ownerFor animals, food is their most precious possession. They'll fight tooth and nail for it. When cats bring their hunted prey home to share with their owners, it's a way of expressing their friendliness and love. Knowing their cats' intentions, wouldn't their owners be moved to tears? For such a profound gift of love, I, for one, am a little overwhelmed—it's just too chilling.
Show off your skillsJust like humans, cats bring home food with a similar feeling. Just as people feel proud when they bring home a certificate of merit as children, cats bring home food to show their owners their prey and let them know that they are also quite capable.
Food reservesMany cats are adopted after living a nomadic life in the wild, often going hungry. To prepare for the future, they store food to prevent starvation. This habit cannot be immediately corrected after adoption, which explains their behavior of bringing mice home.