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1.
"Feed the Birds" was a beautiful song in the musical/movie Mary Poppins, a famous 1964 Disney film. It is about a poor old lady who spends her time trying to feed the poor birds near a cathedral. Meanwhile next door, there is a bank full of people who couldn't care less.

Not made up by Nathan Fillion. Firefly was still a good show though.
But Mary Poppins is beautiful in a way unsurpassed.
Come feed the birds,

Show them you care

And you'll be glad if you do

Their young ones are hungry

Their nests are so bare
by DavidlikesDisney February 25, 2012
 
2.
Not quite fisting someone, where you have all 4 fingers and thumb inserted, but are only able to achieve penetration up to the base of the fingers.

Etymology: Put your fingers and thumb together so they all meet at the tips. Include a back-and-forth motion from the wrist, and your hand looks like a bird eating.
"Brah, I took that chick home last night."
"Oh yeah, did you get laid?"
"Nah. I did feed the bird, though."
by Slartoox February 19, 2010
 
3.
Nathan Fillion, a brilliant actor, coined this famously popular original phrase. "Feed the birds" is used as a way to congratulate, or give props to someone. Here is its meaning as he describes it on his Twitter page:

"This is the new High Five. It is called: FEED THE BIRDS."

Nathan Fillion, the extraordinarily talented, hot, handsome star of "Firefly", "Serenity", and "Castle", just to mention a few of the fine roles of this Emmy-winning performer, also says "Feed the birds" in his show "Castle".

Its short form is FTB
Here are some of Nathan Fillion's quotes as examples:

Example #1:

Nathan Fillion: Just did a Feed the Birds in episode three of Castle. Also, tried to squeeze in obscure Firefly reference. Bam, said the lady.

Example #2:

Nathan Fillion: "First, an Emmy, dinner with friends, wrap it up with @thedanband. Let's call it a perfect evening. Feed the birds."

Example #3:

Nathan Fillion: "Interesting. Guest star was supposed to call me Castle, but instead called me Captain. Feed the birds."

Example #4:

Nathan Fillion: "I saw so many of you have donated to the Children's Hospital Triathlon... I'm thrilled. And touched. Thanks. Really, really thanks... you. Feed the birds, all of you. Extra feed. Extra birds."

Example #5:

Query: When you say "Feed the birds", are you the feeder or the fed?

Nathan Fillion: "The person getting props gets fed."
by Kcat10 July 16, 2010