Legolas: I can not go back.
Thranduil: Where will you go?
Legolas: I do not know.
Thranduil: Go to the North. There is a member of the Dunedain you should meet and stay with. His father was a good man, I have hope he will be a great one.
Legolas: What is his name?
Thranduil: He is known as Strider. His true name, you must discover for yourself.
Kili: I will not hide, when others fight our battles for us!
Thranduil: So this is the Halfling who ate my food, and stole keys from my guard.
Bilbo Baggins: Yes. Sorry about that.
Thorin Oakenshield: Farewell, Master Burglar. Go back to your books, your fireplace. Plant your trees, watch them grow. If more of us valued home above gold, it would be a merrier world.
Bilbo Baggins: One day I'll remember. Remember everything that happened: the good, the bad, those who survived... And those that did not.
Answer: When Aragorn was 25 he left Rivendell to travel the world, having had his proposal to marry Arwen rejected by Elrond. Hence he is absent during the time period of the Hobbit, having gone to Rohan and then Gondor in disguise. During his travels he becomes very well known among the Elves, owing to Elrond raising him and Galadriel welcoming him to Lothlorien after he comes back from his long travels. Even before, he had served in the Rangers of the North as he was growing up, where he gained the nickname Strider. Legolas already knows Aragorn before LoTR, as Aragorn had initially left Gollum in the care of the wood-Elves before someone helped Gollum escape.