I understand that Captain Archer's hand skills saved his life and the lives of others more than once. I also understand that Captain Archer had to frequently time travel, due to the temporal cold war (which wasn't really all that cold). However, I am confused as to why Captain Archer competed in boxing while traveling back to 20th century Earth?
That image isn't from Star Trek Enterprise; it's from Quantum Leap, an unrelated sci-fi series also starring Scott Bakula (who played Archer on Enterprise).
Specifically, the image is from "The Right Hand of God", the third episode of season 1; this fansite has collected more images from the episode, including a few very similar to the one in the question.
The premise of Quantum Leap is that Bakula's character has his consciousness sent through time, where it inhabits various historical personages. In "The Right Hand of God", he inhabits the body of a Sacramento boxer in 1974, where he has to win at boxing match to help some nuns build a new chapel.