I'd just like to add to all the other wonderful answers. We have to consider that the people involved in the Fellowship are very "high born".
The Fellowship consists of
- a Wizard
- a (future) High King
- an Elven Prince
- a descendent of Durin the Deathless
- a future Steward of Gondor
- a descendant of the Shire Thain
- a future Master of Buckland.
Frodo should be considered as Hobbit gentry - he is high born relative to his people, and Sam is his devoted servant. The only one who isn't of relative "high" birth is Sam, yet he arguably is the most devoted of the entire Fellowship, although I think for reasons slightly different than the others.
Being that these members of the Fellowship are of such high birth, it doesn't seem odd that they would be so supportive to the cause of the ring-bearer and the cause of Middle Earth. Aragorn, being raised partially in Imladris, knows this history of his people and much of Middle Earth. He knows of Morgoth, of Sauron, and the evils they have wrought upon Arda. Therefore it is in his best interest to support the ring-bearer and, if I might put forth some supposition, views Frodo to be of high honor for his self sacrifice for the greater good. Consider what Frodo did when he learned of the danger to the Shire and the world at large : he sold his beloved estate and set off, seemingly alone, into parts unknown. While Frodo hoped to only need take the ring to Imladris, part of him knew that it wouldn't stop there, and he went anyway. That deserves some respect, and Aragorn, being educated and learned in history and honor due to his lineage, knows this - I believe he even remarks on Frodos courage several times (I will find the appropriate quotes when I can get to my book). Continuing my supposition, I would say most of the Fellowships knows this at the beginning, or figures it out pretty quickly.
This doesn't bring into account the friendship that develops between the Fellowship and just how strong that bond is.