Why say 少说两句 to break up a fight?

This phrase is similar to "take a step back" in this context. Since most of the stuff said during an argument like this would be provocative (as you can tell from the movie), asking people to "say fewer words" helps both sides to calm down, at least that's the intention.

少 - less

说 - say

两句 - literally means two sentences - actually means several sentences

少说两句 - say several sentences less (than what you are going to say)

n.b. In Chinese language numbers sometimes refer to a general idea of quantity to express many or few rather than a precise figure.

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Thanks for the kind reminding by blackgreen. Here are two more examples below.

两 means a few rather than two:

When you are supposed to finish your speech but actually you need a bit more time, you can say "抱歉，我再多说两句", which mean "excuse me, I would like to say a few more sentences" rather than literally "two more sentences". Hope that would be helpful!

千 means many and much rather than thousand:

Another example of the general idea of "many" expressed by some big words like 百,千,万. There is a sentence in an ancient poem "桃花潭水深千尺，不及汪伦送我情". The general translation would be, the water of Peach Blossom Lake is of thousands of 尺's depth(尺 means ruler, an ancient unit of measure), but it's still not as deep as the friendship that my friend 汪伦 gives me. Here 千 doesn't mean exactly how many thousands. It's doesn't matter. It is used to express the water as well as the emotion is very "deep".

少说两句 is a milder way to say 别说了 or 闭嘴(stop talking).

It can be used whenever you want people to stop speaking/arguing/complaining/etc. E. g. 行了，你少说两句吧。

@dROOOze has already pretty much answered this, I just want to share a reference:

Grand Ricci

少说两句
shǎo shuō liǎng jù
Ne parlez pas tant!

Google Translate renders this into English as:

Don't talk so much!