Vronsky asks Dolly to speak with him. He wants to legalize his arrangement with Anna; if they have more children, they are technically still Karenin’s now, but Vronsky sees himself and Anna as bound together. Anna needs to write a letter to Karenin requesting a divorce, which she hasn’t done yet. Dolly promises to talk to Anna.
Vronsky wants to make his relationship with Anna legal so that they can start to build an actual life together, rather than simply float in the neither-here-nor-there world of an affair. Anna has been avoiding the hard truth of facing Karenin to finalize the divorce, instead preferring to ignore reality and stay in her dream world.