Why is Winston's memory of the Two Minutes Hate important for him to write down?

It's important for Winston to write down his experience of the Two Minutes Hate, because that makes it real. A subtext of the novel is that what's written down is real. Winston keeps a diary so that his thoughts and his criticism of the party become real, both to him and to others who may come after him and read his words. He hides his diary because he knows he's created proof of his disobedience, which could result in his arrest and execution. When he writes down his experience of the Two Minutes Hate, he creates proof that Big Brother is not all powerful.
The political scientist James C. Scott called these types of actions "the weapons of the weak" in his book by the same name. According to Scott, people who are living under all-powerful dictators or governments can regain a sense of their own power by saying or doing small things which contravene official party doctrine or even break the law. These things usually escape the notice of the authorities, because they're small. However, they have an energizing effect on the weak people who use them, because they've proved to themselves, and to others who know their secret, that the all-powerful figure isn't so powerful after all.
This is perhaps the most important reason Winston writes down his memories. They are the basis of his rebellion. He feels powerful against Big Brother because he writes down prohibited and proscribed thoughts. This makes his next step—falling in love with Julia—possible. This heretical act makes possible his last, biggest act of rebellion: the accepting of O'Brien's invitation to join the Brotherhood. Note that it's not Winston's rebellious acts that eventually cause his demise, but O'Brien's treachery. That's a subtle clue that it's still possible for others to follow in Winston's footsteps.
https://yalebooks.yale.edu/book/9780300036411/weapons-weak