Byron was apparently kissing his own reflection in the sideview mirror of the Brown Bomber, the Watson's 1948 Plymouth. Since it was a bitterly cold day, well below freezing, his lips froze onto the mirror when they made contact.
The furnace at the Watson home was not working, so Dad called Aunt Cydney and she told the family to come right over to her house. Unfortunately, the Brown Bomber's windows were frozen solid, so Byron and Kenny were assigned the chore of scraping the ice off so the family could make the trip to Aunt Cydney's. Unhappy at being asked to work, Byron was pouting, and while Kenny was struggling with his share of the ice, Byron was goofing around. His friend Buphead had stopped by and was egging Byron on, and it was somewhere in the midst of their shenanigans that Byron admired his own reflection in the mirror, gave it a kiss, and got stuck.
Byron is going through a difficult stage of adolescence. He is rebellious, highly susceptible to peer pressure, and very concerned with his looks. It is important to Byron that he look and act "cool" at all times. To his chagrin, he ends up looking just the opposite when he gets his lips stuck to the mirror (Chapter 1).
Byron and Kenny were sent by their father to scrape the ice off of the car windows so they could go to their Aunt Cydney's house. So, while Kenny was scraping the passenger's side of the Brown Bomber, Byron was busy admiring his reflection and trying to kiss himself. So, in the process, Byron's lips get stuck to the mirror. When Mrs. Watson finds out, she yanks his lips off of the driver's side mirror.