Brief Summary
The Setup
George Washington decided that he was definitely going to step down from being president at the end of his term. During his announcement of this decision, he thought he should give some advice about politics based on what had been happening in his administration over the previous few years.
The Text
First, Washington gets through the main point of the farewell address: telling people that he's taking himself out of the option pool for president in the upcoming election period. There's also some nice stuff about how he's really not that great and his only successes were because of the American people and their support.
Aww, shucks.
Then, Washington goes on to spend some quality time talking about why Americans should try to avoid letting themselves be divided into factions over politics. They'll be a lot stronger when united, but when party politics take over, they run the risk of a despot taking over. Although he doesn't say it explicitly, this is very obviously a reaction against the rise of political parties during his presidency.
The other big piece of advice Washington dispenses is that the United States shouldn't get too close to any other foreign countries. This just leads to forced commitment to (and forced antagonism with) another country, which means potentially fighting for or against things that the United States doesn't really want to be involved with.
TL;DR
Washington wants to tell the world he's done being the first president—also, while he's got their attention, Americans should stop letting political parties tear them apart and picking their favorite foreign countries. Easy-peasy.