So, the Ra'Zac show up in Carvahall at the beginning of Eragon. They don't know where the 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:
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
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.