So, the Ra'zac show up in Carvahall at the beginning of Eragon. They don't know where the dragon egg is, they are really just shooting in the dark looking around. It says in the book that the Ra'zac were following Brom, but it seems unlikely that they knew where he was and did nothing about it for 15 years... How did they know to show up there?
It's also discussed further in the books, than mentioned in BenjaminHarris's answer on pg. 226 when Brom and Eragon are leaving Teirm, Eragon demands that Brom Give him some more answers.
When Eragon asks how Saphira's egg appeared in the Spine, the follow passage occurs:
"No, Saphira was taken from the Varden, and I think I know how. To protect the egg, its guardian must have tried to send it to me with magic."
"The Varden haven't contacted me to explain how they lost the egg, so I suspect that their runners were intercepted by the Empire and the Ra'zac were sent in their place. I'm sure they were quite eager to find me, as I've managed to foil many of their plans."
"Then the Ra'zac didn't know about me when they arrived in Carvahall" said Eragon with wonder. "That's right," replied Brom. "If that ass Sloan had kept his mouth shut, they might not have found out about you.
So The Ra'zac don't arrive in Carvahall looking for Brom. As to how they found Brom or the runners Brom suspects would have been sent by the Varden, while in Teirm, there is also talk of Traitor(s) in the Varden that are the cause of the goods of the Merchants sympathetic to the Varden going missing.
It is never explained in any of the books in the Inheritance Cycle. The only time it is discussed is near the beginning of Eragon, when Brom says the Ra'zac were in Galbatorix's service. Eragon asks Brom how Galbatorix could have known about Saphira since nobody saw her, and Brom concludes that
The Ra'zac must have had a source of information outside of [Carvahall], one who is probably dead... (Eragon, p. 104)
It's likely that the Ra'zac happened upon Carvahall by accident and were alerted by mention of Eragon's stone. Sloan told the Ra'zac about it, but others like Horst also knew. Meeting Brom prowling near their camp would have confirmed their suspicions.
The Razac must have had a "score of information outside Carvahall", like Benjaminrh mentioned.
The outside of Carvahall info score could have been that trader that Eragon and his uncle tried to sell "the stone" to. Durza would have alerted King G. that the egg was loose, and they would have been listening for anything that had to do with it, so if the trader had ever mentioned it...
And he probably would have, too, with it being such an odd artefact, he'd want to find out what it could have been.
So, the Razac came, and Sloan told them where to look.