Groom has two main meanings:

1. To clean and care for appearance. When you groom yourself, you brush your hair, wash your face, and make sure you look neat and presentable. Dogs and cats groom themselves by licking their fur clean. Horse owners spend time grooming their horses with special brushes to keep their coats shiny and healthy. A well-groomed person pays attention to details like clean fingernails, combed hair, and tidy clothes.
2. A man who is getting married or has just gotten married. At a wedding, the groom is the man standing at the altar waiting for his bride. People often say “the bride and groom” when talking about the couple getting married. After the ceremony, he might still be called the groom for a while, like when people talk about “the groom's family” or “the groom's speech” at the reception.

The verb form matches the first meaning: you can groom a dog or groom yourself by cleaning and caring for your appearance.