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.
Quote #4
[…] for the efficient management of your common interests, in a country so extensive as ours, a government of as much vigor as is consistent with the perfect security of liberty is indispensable. Liberty itself will find in such a government, with powers properly distributed and adjusted, its surest guardian. (19.3-4)
Again, Washington tries to convince the audience that the federal government, despite what Anti-Federalists say, is the best way to protect people's freedom. Fear over the development of a centralized government was largely based at the time on the former colonists' experiences under a monarchy, but Washington repeatedly tells them that the federal government will be nothing like that.
Quote #5
[…] nothing is more essential than that permanent, inveterate antipathies against particular nations, and passionate attachments for others, should be excluded […]. The nation which indulges towards another a habitual hatred or a habitual fondness is in some degree a slave. It is a slave to its animosity or to its affection, either of which is sufficient to lead it astray from its duty and its interest. (32.1-3)
Another of Washington's big ideas in the farewell address is "don't make alliances with other countries." The big reason, as he mentions here, is that when you do that, you bind yourself to that other country and ruin relationships with others. It takes away from your ability to make your own decisions. In other words, it takes away a country's freedom to act in its own interest.