I am assuming this is a sincere question, so I'll give a sincere answer.
Peterson has consistently made two arguments (that come to mind) that cause these accusations:
- his arguments that "dominance hierarchies" naturally favour men, as they are more aggressive, more disposable, more risk-taking and less burdened by the necessities of child-rearing, and are not an invention of "the patriarchy". He supports this argument by pointing out its near-universal equivalent in nature, especially lobsters. These structural realities inevitably hinder women's achievements in the workforce, which he suggests is purely natural.
- He notes that families with two parents are, on the whole, more stable and successful than families with only one, and suggests that society should make it less easy to create so many single-parent families, for example by placing limits on divorce.
Both of these arguments make a, to some, very unwelcome point - that perhaps gains in women's rights have, and will, come at a huge cost to society that is only going to get worse, and that we're a couple of decades into a giant experiment that, for a lot of people, could turn out very badly indeed.
This kind of thing is simply unsayable in today's climate and, not surprisingly, he's public enemy #1 in some circles. That said, I don't think his ideas are merely "misogynistic" and he just hates women. Needless to say, however, some of these ideas have been seized upon by actual misogynists to justify their ludicrous beliefs, which rightly or wrongly has not exactly helped his cause in the eyes of the twitter mob.
For what it's worth, I both agree and disagree. I do think single-parent families are a real problem, and I agree that the necessities of childrearing structurally hold women back from their full career potentials. I don't, however, agree that the solution is to roll back divorce laws, or send women back to the kitchen (not that he said this, but he hasn't said NOT THIS!). Perhaps society can offer (much) more child-raising support. Maybe we need to retreat from the idea of the nuclear family altogether, and raise children in extended families or even communities (there are examples of this in different cultures). We need to do something - but I can't stomach the idea that we can just go back to the "good old days (for some)".
He's right about many things, but he's also curiously stuck in the past, and very boxed-in by his own personal experiences - which has been a double edged sword. We can't simply roll back people's rights in order to solve a problem, whether that problem is real or not! His real blindness is not being able to offer palatable solutions, or to understand why people might not like the ones he has made.
Anyway, that's the (perhaps unfair but anyway highly unsurprising) reason people think he's misogynistic.