This is Washington's farewell address, after all, so it's probably good for him to mention why he's saying farewell. Washington begins and ends his address by talking about his retirement to his farm.

And by "farm," we mean "large plantation."

Emphasizing that he's going to retire has two main effects on the address: first, it gives Washington a concrete excuse for leaving. Second, it makes the decision seem more final.

It's not like he's leaving because he's lazy or got a better offer from another company. In fact, the way he puts it:

I constantly hoped that it would have been much earlier in my power […] to return to that retirement from which I had been reluctantly drawn […] but mature reflection on […] our affairs with foreign nations, and the unanimous advice of persons entitled to my confidence, impelled me to abandon the idea. (3.2)

The retirement seems even more legitimate since Washington got pulled out of it in the first place and has been kept from it out of a sense of duty. Most people can see the appeal of being retired, especially when you're a war hero, so the image of giving that up could be pretty appealing.

At the end of the address, he gets a little more indirect (and morbid) about the idea. He hopes that "after forty five years of my life dedicated to its service with an upright zeal, the faults of incompetent abilities will be consigned to oblivion, as myself must soon be to the mansions of rest" (49.3). That last part seems like it's about dying, which is not quite the same as retiring, but the implication is that he's thinking about the final countdown, so some time to relax beforehand might be nice.

Of course, he didn't know that he was going to die in a couple years' time (unless he had some abilities we don't know about), but it seems like he's trying to really make this decision seem final.

If someone leaves to go to a different job or move to the country or something, their colleagues will try to convince them to stay. Assuming they're good at their job, of course. If they're retiring (and pondering their death), that's a lot harder to argue with.