Bilbo is the kind of person that bottles his pain inside.
He was witness of Fili´s final moment and while the others started reacting (with rage, tears) he just looks back at where Fili was, almost not wanting to believe what he saw.
After Thorin´s passing we see him with Gandalf, almost immobile, looking ahead, and then he tries this little smile. Later on, while trying to speak about Thorin with Balin, his eyes start to water and he prefers to walk
away before he could say more. And finally, after he arrives at Bag End and he´s forced to talk about Thorin, he quickly gets inside. The only chance he allowed himself to show his grief, was while being by Thorin´s side. Even though he knew the dwarf´s soul was not there anymore, his only presence was all it took so he could show his feelings.
The most heartbreaking part is that Bilbo will have to endure many years like this, on his own. Not being able to reveal his sorrow to nobody else.