That secret something Anna has been hiding from Vronsky is that she wants to see her son. That was the main point of her desire to visit Petersburg.
For two days, Anna has staked out her objective, making inquiries and doing recon. (She had ruled out going directly to the house or appealing to Karenin.)
On the third day, Anna discovers that Karenin and Countess Lydia are close friends. She writes the Countess a note (the note that we see in Chapter 23), knowing that Karenin will live up to the magnanimous role he is playing.
Anna feels utterly humiliated when "no answer" comes back, but she sees and understands Countess Lydia's point of view. What makes it worse is that, since she hasn't confided in Vronsky about wanting to see her son, she has to suffer alone.
The next day Anna is composing an appeal to her husband when Countess Lydia's letter is brought to her. It makes her angry due to its clear spitefulness and cruelty.
Anna resolves to go to the house tomorrow, on Seryozha's birthday, and do whatever it takes to see him—i.e., bribe the servants.
She buys a lot of toys and gets ready to go to the house before Karenin is awake. She's prepared for everything except what to say to her son.
The next day, Anna goes to her old house. Kapitonych recognizes her and lets her in. She goes straight to her son's room.
Anna is amazed at how much her son has changed physically. She starts crying.
The reunion between mother and son is joyful. Seryozha exclaims that he knew she would come because it is his birthday.