The questioner is right to be confused. Henry, in his attempt to inspire his men to battle, seems to use the wrong word: "ne'er" instead of "e'er". The king is saying, I think, that even the most vile, meanest, lowest of society, will be "gentled" or raised to a higher class by taking part in this selfless act of battle. But for that sense, Henry should say, "Be he e'er so vile, this day will gentle his condition...." I have pondered this sentence occasionally for years. Perhaps he uses ne'er in the sense that the low will no longer be vile, they will be gentled, so the sentence loses its contradictory sense of before and after--which seems to me to be a weaker statement. Perhaps he could have inserted "as" instead of the comma, or perhaps "as" is understood even though unspoken.
When understanding a piece of literature, it helps to research the background of the piece. Your particular quote, "be he ne'er so vile,/This day shall gentle his condition," comes from Henry V by William Shakespeare. This is the speech for St. Crispin's Day. So, you may ask, "Why is there a St. Crispin's Day?" A blogger for CEOsherpa summarizes the speech, "Shakespeare created a rousing motivational speech in which Harry the King exhorts his tired English troops on the morning before this historic battle." This is a speech for encouragement!
To answer your question, "be he ne'er so vile," means that King Henry is never vile, or repulsive. "This day shall gentle his condition," means that the day changes King Henry.
Your next question for better understanding might be, "How does the day change King Henry?"