Summarize the following:
She will tell you to kill one of the captives. You can kill one, two, or all of them.  You can speak with the captives and listen to their stories. Kill whomever you think deserves to die. Your decision has no impact on gameplay. She will congratulate you and comment on your decision of which captive(s) to kill, and then instruct you to meet her at the Dark Brotherhood Sanctuary. It's in the Southwestern part of the map just west of Falkreath. It's the door with a skull on it.  This is the entrance to the Dark Brotherhood Sanctuary.  The door will ask you a riddle; "What is the music of life?". This is the correct answer to the riddle.  It's the first option in the answer menu.  The Black Door will respond, "Welcome home" then you will be let into the Sanctuary. Once you are inside, locate Astrid and talk to her. You will be made a full member of the Dark Brotherhood. You can now take on killing contracts for money (usually a few hundred gold).

Summary:
Talk to Astrid. Speak with Astrid. Go to the Sanctuary. Activate the Black Door. Select "Silence, my brother" as the answer. Talk to Astrid.