Washington breaks it down pretty clearly:
1) I'm outta here (thank goodness).
2) Don't let political parties be a thing.
3) Don't pick favorites when it comes to other countries.
This speech represents Washington's parting words of advice to an audience he's not really sure is going to listen. He claims that he's not that great, but he really tried. Then, he tries to get people to behave morally in politics instead of letting a lot of infighting cause problems.
Well, you can't say he didn't warn…everyone.
Washington made sure to squeeze in a lot of advice in his goodbye speech because he really isn't confident about the direction the country is going.
Washington published the speech instead of delivering it to Congress because the more people you spread advice to, the better the likelihood that someone will listen to you.
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.
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.
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.
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.