After obtaining secret documents from the Thalmor Embassy in the quest "Diplomatic Immunity", you learn that the Thalmor are also seeking information about the return of the dragons. The intelligence you gathered from the embassy point toward a Blades member called Esbern, who is located in Riften. Delphine suggests that when you arrive in Riften, you make contact with Brynjolf, who is well connected and should know the whereabouts of Esbern.
Open your quest journal:A Cornered Rat
The Thalmor are just as puzzled as we are about why dragons are returning. I learned that they are looking for a Blade named Esbern, who Delphine says is an expert on ancient dragon lore. We need to find him before the Thalmor do.
Talk to Brynjolf
Riften is located to the east of Ivarstead. It is pretty far out, so if you have not been there and the possibility of fast travel does not exist, it is recommended you take a carriage from a town such as Whiterun or Solitude.
When you arrive at Riften, the guards will not let you in. They want you to pay a visitor's tax. You can either PERSUADE the guards that it is obviously a shakedown, PAY the visitors tax, or INTIMIDATE them saying that you kill thieves. If your Speech skill is high enough, try to persuade or intimidate them. They will unlock the door for you for free. Otherwise, pay the tax. Enter Riften.
Near the entrance, a goon name Maul will offer to give you information in return for some gold. You can pay him up or try to PERSUADE him to give you the information for free. You can learn about the thief guild, and that the contact you are looking for, Brynjolf, can be found in the market.
Find Brynjolf, in the market, tell him you seek information. He will want you to do a small task for him first. This starts the Thieves Guild Quest "A Chance Arrangement". This will also open up membership in the thieves guild and unlock a whole new series of quests.
This quest is covered in our Thieves Guild Series Walkthroughs. Refer to the walkthrough before continuing: A chance Arrangement Walkthrough
If you have already done A Chance Arrangement previously as part of applying to the Thieves Guild, You can instead just go talk to Brynjolf in the Ragged Flagon and he will be happy to give you the location of Esbern.
Brynjolf tells you that Esbern has been hiding in the Ratway Warrens. Climb down the stairs NW of the market, and enter the Ratways. Work your way in the tunnels, heading to the east, enter the Ragged Flagon, then exit through the door to the Ratway vaults. The Thalmor, who are furious about the Embassy incident, have sent soldiers after you. So be on the watch for Thalmor Soldiers. Work your way east. In a room with three gates, drop to the lower level, go through the gate, and head south. Enter the Ratway Warrens.
Go up to the second floor and open a door to the south. Watch for the crazy people in this area, Hefid the Deaf and Knjakr the chef who can get violent if you get too close to them. The shutter on the door will slide open. Say you are looking for Esbern, the man behind the door will tell you to go away, he doesn't know who Esbern is. Tell him it is ok, Delphine sent you. Persuade him that Delphine needs his help. He will finally let you in. If the persuade doesn't work, you can either tell him Delphine's quote about remembering the 30th of Frostfall, or tell him you are a Dragonborn.
He will tell you Alduin has returned, just like the prophecy said. That the world will come to an end. That the situation is hopeless since nothing can stop Alduin except a Dragonborn, and there hasn't been a Dragonborn in ages. Tell Esbern that you are a Dragonborn. He will regain faith and see that the situation isn't hopeless anymore. He wants you to take him to Delphine immediately.
This completes "A Cornered Rat" and starts the next quest of the main story campaign, "Alduin's Wall".