There isn't any one reason why everyone agreed to let Frodo carry the ring; each person has their own reasons. None of the council members ever explicitly say why they agreed to go with Frodo, but we can infer some of their individual reasons by their reactions:
For starters, I should point out that Frodo had a lot of qualifications that made him logically the best candidate to carry the ring, In particular:
- Gandalf and Elrond had already commented how Frodo, and Bilbo before him, seemed unusually resistant to the ring's influence.
- Frodo was already a ring bearer, and it was certainly a good idea to limit the number of people that had direct contact with it.
- Frodo was "just a hobbit"; even if he did succumb to the ring's influence, his capacity to cause real harm would be limited.
Beyond that, Gandalf certainly seemed to believe that Frodo was destined to be the one to carry the ring; when Frodo finally did volunteer, Gandalf's face was not surprised, but resigned. Gandalf's opinion would also have likely swayed Aragorn, on top of the fact that Aragorn had already traveled with Frodo, and "knew his heart".
Gimli and Legolas likely agreed to let Frodo carry it primarily because it meant keeping it away from the other. Legolas (who agreed first) was also likely swayed, at least a bit, by Gandalf and Aragorn's confidence, since he held both of them in high respect. Gimli would have then had to agree just to avoid being seen as cowardly in the face of the Elf.
It's interesting to note that Boromir doesn't actually seem to agree with the decision to let Frodo carry the ring, but he is resigned to accept "the word of the council". From subsequent events, we know his only motivation is to get the ring back to Gondor. He probably saw Frodo as an easy mark, and decided that the best way to get the ring for himself was to follow along with everyone else and wait for an opportunity to steal it.