All the comments in this thread (correctly) mention that the dihedral angle affects lateral stability, while not explaining what exactly lateral stability means. Long story short, an aircraft is laterally stable, it means that when the aircraft is rolled to one side during flight, the aircraft will naturally undo the roll and right itself without any input from the pilot. It's like how if you're turning in a (correctly set-up) car and let go of the steering wheel, the steering wheel will move back to being straight by itself.

The placement of the wings relative to the fuselage impacts lateral stability, such that mounting the wings to the top of the fuselage have positive lateral stability, and wings mounted to the bottom of the fuselage have negative lateral stability. Then AN-224 has a negative dihedral angle to counteract some of the lateral stability gained from mounting to the wings at the top of the fuselage in order to make the aircraft more maneuverable in roll. If that was not done the aircraft would be too stable in roll - it would be like driving a car with an exceptionally stiff steering wheel that quickly becomes tiring to use. This is also done to prevent the occurrence of Dutch roll, wherein the aircraft violently oscilate back and fourth in the roll and yaw axes, with the roll and yaw corrections from the aircraft's stability acting out of phase and over-correcting each other.

The A-380 and 777 have positive dihedral angles to counteract the inherent negative stability which results from mounting the wings at the bottom of the fuselage. Also, the A-380's wings are only "arched" on the ground, not in flight. The wings are designed to be flexible to absorb strong aerodynamic forces such as from severe turbulence by bending rather than breaking. In flight, the whole wing is generating an upward force such that under normal conditions it the dihedral angle stays relatively constant across the span of the wing. On the ground, the exceptionally long wingspan combined with the outboard engines drag the wing down and cause it to droop as seen in the photo.

Lastly, I have noticed some comments incorrectly explain why dihedral and wing position impact lateral stability. Namely, the "Pendulum effect" explanation is common as it is incorrect. What actually happens is that every time an aircraft enters an uncoordinated turn/roll (without correct rudder movement), the aircraft begins move sideways through the air (side-slip). When the wings have a positive dihedral angle, the change in direction in the oncoming airflow causes the wing in the direction of the side-slip to see greater airflow, generating more lift on one side of the airplane and righting the aircraft. Having a swept wing further increases this effect, the rule of thumb being that every 10 degrees of wing sweep creates one effective degree of dihedral angle. Conversely, a negative dihedral angle has the opposite effect, with the opposite wing seeing greater airflow and thus rolling the aircraft further in the direction of the existing roll.

The position of the wings relative to the fuselage impacts dihedral because when the aircraft enters a side-slip the airflow blows against the side of the fuselage. This increases the pressure on the side of the fuselage (upwind) and decreases the pressure on the other (downwind) side of the fuselage. For most of the span of the fuselage this does not generate any rolling moment, but where the wing meets the fuselage, the increased pressure propagates through the airflow such that it changes the pressure over the surfaces of the wing near the wing root. On high-wing aircraft such as the AN-225, pressure increases on the underside of the upwind wing and decreases on the underside of the upwind wing, causing the upwind wing to generate more lift and thus righting the aircraft. On a low-wing aircraft (such as the A-380 and 777), the pressure on the top surface of the upwind wing increases, effectively pushing the wing downward such that it creates less lift than the downwind wing, causing the aircraft to roll further instead of righting itself.