How we cite our quotes: (Paragraph.Sentence)
Quote #1
Interwoven as is the love of liberty with every ligament of your hearts, no recommendation of mine is necessary to fortify or confirm the attachment. (8.1)
Not only is this quote pretty poetic, it's actually pretty anatomically accurate. Washington is also reminding his audience of their own feelings toward freedom (a.k.a. "liberty") to set them up for his later advice about government and foreign policy.
Quote #2
Hence, likewise, they will avoid the necessity of those overgrown military establishments which, under any form of government, are inauspicious to liberty, and which are to be regarded as particularly hostile to republican liberty. In this sense it is that your union ought to be considered as a main prop of your liberty, and that the love of the one ought to endear to you the preservation of the other. (13.2-3)
Tyrannical governments tend to use military force to get and keep their power. It's the kind of power play the American colonists were trying to get away from in the Revolutionary War. Washington is advising everyone to avoid that kind of system in the United States by supporting the work of the federal government. The government will protect people's freedom, in theory.
Quote #3
[…] they are likely, in the course of time and things, to become potent engines, by which cunning, ambitious, and unprincipled men will be enabled to subvert the power of the people and to usurp for themselves the reins of government, destroying afterwards the very engines which have lifted them to unjust dominion. (18.1)
The "they" in this scenario are political parties, which Washington sees as a seriously negative development in U.S. politics. This quote explains why, at least in part. If you give a mouse a political faction, he'll use political discord to eventually seize control and take away his enemies' freedom.