The author joined a women-dominatedwent to a predominantly female party, but one man was particularly talkative, while his wife kept silent.

"Join a party" suggests political party. "Go to a party" or "joined his/her friends at a party" would be preferable.

At the end of party, the man claimed that at home, his wife iwas the talker at home and he hasd nothing to say.

Since both actions ("his wife was the talker" and "he had nothing to say") take place at home, this word order is more suitable. The original word order sounded ambiguous (nothing to say at that moment, or at home?). It may have been clear from context which interpretation is correct, but it still didn't quite sound right.

The present tense would be acceptable to some people, but for tense matching, I think the past tense is more appropriate.

The irony does exist that while men tend to talk more than women do in public situations, they often talk less at home.

"That" only introduces one idea, so you need an appropriate conjunction to combine both parts ("men tend to talk more…" and "they often talk…").

This pattern is wreaking havoc withon marriages because most wives want their husbands to be conversational partners, but few husbands share this expectation of their wives.