What is the initial conflict between Roy and Beatrice Leap? How do their interactions move the plot along?

It's fair to say that Roy and Beatrice get off on the wrong foot. One morning on the school bus, Roy pushes past Beatrice, “a tall girl with curly blond hair and red-framed eyeglasses” to get off so he can follow Mullet Fingers. As Roy soon discovers, this is a big mistake. Beatrice has a well-deserved reputation for being incredibly tough, a girl with attitude who won't take any nonsense from anyone. Roy learns this for himself later on at school when Beatrice confronts him over the incident on the bus. This is a girl who clearly stands up for herself. She's also fiercely protective of Mullet Fingers, who just happens to be her stepbrother. So the source of the initial conflict between Roy and Beatrice is twofold: Roy's rudely pushing her out the way on the bus, and Beatrice's determination to protect Mullet Fingers from what she perceives as a threat.
The initial conflict is important to the plot as it establishes that we're dealing with two strong characters who are fiercely loyal and protective of the people and things they really care about. This shared attitude will allow Roy and Beatrice to overcome their initial animosity and join forces to save the owls' habitat from being destroyed.