I watched this video a few times, to understand exactly what the connection between these two characters really were. I came to this conclusion:
The Ice King and Marceline are one of the most emotionally affected characters through out the adventure time series, due to the mushroom war. If you have seen the episode, you have heard the notes that the Ice King had written to Marceline, and you saw the ending clip, of which the Ice King, (Or Simon) gives Marceline a new best friend, Hambo. I was curious of why Marceline had such strong feeling for such a minor happening, but after a while I thought I began to understand.
The two characters obviously had a past before that clip, a strong relationship, almost like best friends. The Ice King's crown was beginning to take him over, as seen in last year's Christmas special. I thought of the crown as some form of drugs, as he had said it was taking him over, and he is losing himself, and he is afraid that Marceline will lose him too. But that clip symbolizes, that after the war, when billions were dead, and just before the Ice King became who he is today, he did what he did. What I felt happened, is the Ice King told her through this, "Don't be afraid, Hambo can take care of you now. I am so sorry I can't look after you, but there is nothing I can do. I love you, and good luck." Marceline held on to the doll until the day that her boyfriend Ash sold it, which left her in tears.
I believe these two characters have a long past, and I feel that this episode was extremely heart warming. Please! Give me your thoughts!