I think what happens when you move from an insecure to a secure attachment is that you are no longer afraid of and for the relationship. The dependency stabilizes and levels off so that you don't feel extremely needy and dependent one day and highly ambivalent or frightened of it the next. You begin to accept the level of dependency that is healthy for a therapeutic relationship, and you can separate fact from fantasy because you can accept and understand that you didn't get what you needed as a child and you don't need to push those feelings away anymore because they are understood and accepted by your T. You can talk freely about what you wanted and needed without feeling ashamed of those needs. You can talk about your fantasies and wishes that your T could be the parent you never had, and as you talk about it you become better able to regulate all the feelings associated with unmet needs because your T understands and doesn't make you feel ashamed or embarrassed about having those fantasies and wishes.
You slowly move past the wishes and fantasies to an acceptance of the boundaries and limits of the therapeutic relationship. You are no longer afraid to tell him certain things about yourself because you are confident in his ability to be there to listen. You trust him when he says he can hear anything you want to tell him and you feel safe enough to show him who you really are. You stop worrying about what he thinks of you every time you tell him something shameful because you believe him when he says that it takes great courage and strength to face your own demons.
And slowly, over time, you become comfortable with your place and position within the relationship, so you aren't threatened by your T's relationship with other people because your own place and position are secure. No one else will "take it away" from you. You understand and accept that each relationship your T has with other people, whether they are his patients, colleagues, friends, or family, is special in its own way because it is unique, and you realize that your own relationship with your T is just as special because it is unique, so you begin to let go of the jealousy and rivalry and desperation that previously characterized the relationship when your own place in your T's life wasn't secure. You don't need to try to make yourself become more important to him than you actually are because you can accept yourself for who you are in his life, and you can accept his role as your T. You don't need to look for ways to become "special" or more important to him than his other patients/clients are because you are comfortable with who YOU are and what YOU bring to the relationship. The need for extra contact in order to be "remembered" between sessions diminishes because you realize the relationship still exists and he doesn't forget about you even though he doesn't think about you as much as you think about him. Your extreme sensitivity to everything he says and does is greatly diminished because his every word and action no longer characterize the totality of your relationship -- he isn't wonderful one day and heartless the next based on what he does or doesn't do. He can make mistakes or refuse to give you something you want and it doesn't feel threatening because you know it doesn't reflect the overall quality of your relationship. You don't need more from him than he can give you because the security of your relationship allows you to feel less needy within the relationship -- you can look beyond it to other relationships because you have a safe base to return to.
It really is a fascinating process that you can only fully appreciate AFTER you've reached a place of secure attachment, because until then you have nothing to compare it to.