How we cite our quotes: (Paragraph.Sentence)
Quote #1
[…] I will only say that I have, with good intentions, contributed towards the organization and administration of the government the best exertions of which a very fallible judgment was capable. Not unconscious in the outset of the inferiority of my qualifications, experience in my own eyes, perhaps still more in the eyes of others, has strengthened the motives to diffidence of myself; and every day the increasing weight of years admonishes me more and more that the shade of retirement is as necessary to me as it will be welcome. (5.1-2)
If you want to retire and other people don't want you to retire, you can always employ some self-deprecation to try and make your case. Washington is basically saying, "Trust me, you guys don't want me to stay." He tries to claim that other people also see his abilities as inferior, which might be a reference to all the division and tension in his administration.
Quote #2
If benefits have resulted to our country from these services, let it always be remembered to your praise […] that under circumstances in which the passions, agitated in every direction, were liable to mislead, amidst appearances sometimes dubious, vicissitudes of fortune often discouraging, in situations in which not unfrequently want of success has countenanced the spirit of criticism, the constancy of your support was the essential prop of the efforts, and a guarantee of the plans by which they were effected. (6.2)
Another way to show your humility is to credit other people for your successes. Like when that group project you did all the work for gets an A+, you say it was a team effort when, really, it wasn't. Instead of taking credit himself, Washington is telling his readers that they, not him, kept the country afloat during all the turmoil of his presidency.
Quote #3
Nor can I forget, as an encouragement to it, your indulgent reception of my sentiments on a former and not dissimilar occasion. (7.4)
Washington is about to launch into the advice section of his farewell address here. But instead of introducing it by listing the reasons why his audience should listen, he instead presents it as them being "indulgent" toward him. As if they are doing him a service, rather than the other way around. It's kind of like buttering them up so that they'll listen to him more, instead of just showering them with sage advice.