The weight is fixed, it is the force on a fixed mass due to a fixed gravitational field strength. What we're looking at here is the reaction force applied by the bird on the ground (or, by Newton's 3rd, by the ground on the bird), i.e. what you'd see on some scales.
The reason the reaction force is changing is because the bird is squatting/launching, which are both dynamic processes, and because they are dynamic, it means the forces acting on the bird aren't necessarily balanced. This means its weight is not necessarily equal to the reaction force.
For example, when the bird starts squatting, it's accelerating downwards which means the resultant force on the bird must be acting downwards, which means the weight is bigger than the reaction force. Since the weight is fixed, this can only happen if the reaction force falls. That is shown by the initial dip on your graph.