Gandalf is missing in an action when Bilbo and the dwarves meet trolls. Why is this important for the development of Bilbo's character?

This episode in Chapter 2 is crucial for Bilbo's character development because it shows that Bilbo actually has a taste for adventure. More importantly, it allows Bilbo to show that, even though he is not a brave warrior, he can still contribute to the Quest for Erebor.
At this early stage in the story, Bilbo still doesn't fully understand why he chose to join the Dwarves on their quest, and part of him thinks it was a very foolish idea. The Dwarves also have no real sense of what Bilbo can contribute. They accepted Bilbo at Gandalf's urging, and, until this point, Gandalf has always been present as a guide and protector.
Investigating the trolls' fire is Bilbo's first official act as burglar. Bilbo is actually quite keen to test his burgling skills. Unfortunately, it goes poorly: Bilbo is caught attempting to steal a money pouch. The Dwarves hear the racket surrounding Bilbo's capture and are also seized when they come to investigate. Although Bilbo is able to escape, he is unable to help his companions. The trolls start preparing to eat the Dwarves, and the situation looks inescapably grim. It seems like Gandalf and his magic are the only things that could rescue them, but the Grey Wizard is nowhere to be seen—or so it seems.
At this point, it looks like all the doubts Bilbo had about his suitability for the Quest are justified. It almost seems like Tolkien wants the reader to wonder, "Wouldn't it have been better if he just stayed safe at Bag End?"
However, at the same time, we are given clues that Bilbo might be cut out for adventuring after all. He was willing, and almost a little excited, to try to burgle the trolls, even without Gandalf there. We see that he also has enough of his wits about him to take advantage of his small size. This shows that Bilbo doesn't completely crumble under pressure and that he recognizes the advantages of his diminutive stature. While he is out of the trolls' sight, he also finds the key to their treasure horde. 
The fact that Bilbo secured the key to the treasure room ends up being very important and demonstrates he can add value to the Quest. In the treasure horde, Gandalf and Thorin discover the powerful High-Elven swords that Elrond will later identify as Glamdring and Orcrist. Bilbo also finds the small dagger that he will name Sting.