I think it's because he know's his Uncle Bilbo is there. ---- Frodo doesn't know where Bilbo is. But if Gandalf tells him he has to take the One Ring to Rivendell where it would be save, before he is found in the Shire, what else should he do?
Because he knows if he stays in the Shire the Ringwraiths will easily find him and kill him.
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Because he wants to help. Which is also why he was sent to Middle-earth in the first place: to help the peoples of Middle-earth in their fight against Sauron.
Aragorn came from Rivendell to wait for Frodo, and Lord Elrond give him instructions where to go. The Elves know what's happening because they have telepathic powers, or from the Palantir.
According to the book:As told in the first volume, The Fellowship of the Ring, Glorfindel is sent by Elrond of Rivendell to help the hobbit Frodo reach Rivendell as he is pursued by the Nazgûl. He sets Frodo on his horse, Asfaloth, and Frodo rides ahead to the other side of the Ford of Bruinen, where he defies his pursuers. He is nearly captured, but Glorfindel, Strider and Frodo's hobbit companions come from behind and drive the Nazgûl into the water, where they are swept away by a wave of water resembling charging horses (an enchantment of Elrond's and Gandalf's). Strider and the hobbits bear torches, but Glorfindel reveals himself as a mighty Elf-lord terrible in his wrath; Frodo sees him as a shining figure.According to the film:Aragorn , but Arwen took Frodo to the Elves to be cured...
Until the end of The Return of the King, when he went to the Undying Lands with the elves and Frodo.
Frodo knows Gandalf because Gandalf was like Bilbo (Frodo's grand-uncle) guardian on Bilbo's adventures. Now, Frodo knows Gandalf through Bilbo.
Frodo was 33 years old when Bilbo adopted him.