Beaver makes fun of a girl with a new hairstyle in front of his friends and hers. Granted, the girl looks like she's heading to a Bingo game at the local church with her old lady hairstyle.
The girl is smart though and retorts right back to him that he looks like a sheepdog with his hair. (ironically, the Beatles would make this sheepdog hairstyle famous in a couple of years, so Beaver was ahead of his time.)
The girl's retort upsets Beaver so much that he gets a complex over his hair and it serves him right. Did he think his comment wouldn't do the same to her? Anyway, Beaver spends money on hair products to make his hair sleek and shiny. He doesn't look like a sheepdog anymore, but now looks like Alfalfa from the Little Rascals.
Once Eddie gets wind of things he tells Beaver to just insult the girl right back and Beaver loves the idea even though it was him who started the insults first. Shirley was merely retorting to his original insult.
When he follows Eddie's instructions the next day at school, Beaver is mean and insensitive hurling insults to the poor girl and she bursts out in tears making Beaver feel like crap for hurting her like that.
He apologizes and learns his lesson, but when you initiate insults like he did don't cry when you get them thrown back at you. The fact that Beaver wanted to get her back for getting him back for what he started was incredibly immature, but at least he learned his lesson in the end and made things well again.
And to be honest, Beaver's hair looked just fine and the Sheepdog insult shouldn't have bothered him. Shirley's hair was pretty ugly and looked like it should've been on a woman 40 years older, but Beaver should've just kept his big old mouth shut when he saw it.