Speaking from his home where he has been recuperating since he left the hospital on Monday, Barrett said he initially thought something was wrong with the baby before quickly realizing the 20-year-old man was agitated.
Barrett, who had a bloodied gash on his face, said the situation got "very, very ugly, very, very quickly."
A 20-year-old man has been arrested, and Chief Deputy District Attorney Kent Lovern said he expects his office to get the case Wednesday to file charges. The grandmother has said the little girl's birth father was arrested for allegedly beating Barrett.
"The response (from the man) was so bizarre that things took off in the wrong direction," Barrett said.
About 100 neighbors and friends gathered and applauded the mayor Wednesday as he approached the microphone with his wife and his niece who made the 911 call.
As he's been recovering, he's gotten calls from lots of well-wishers, including President Barack Obama. He said he was happy to get the call - but told the president he'd prefer "to be called for pitching a perfect game" instead.