To explain why these speeds change with weight:

V_A design maneuvering speed: this is because it takes a lot more force to move a heavy plane than it does a lighter plane. On a lighter plane, the surfaces deflect, apply a force, and the plane moves. Once the plane is moving, the surfaces move too, which lessens the force on them. A heavier plane will move less, which means the surfaces are applying more force and for longer to move the plane the same amount.

V_X best angle of climb speed
V_Y best rate of climb speed: A heavier plane will need to produce more lift, and therefore have more drag, which affects climb performance.

V_R rotation speed
V_S0 landing configuration stall speed
V_S1 "specific" configuration stall speed: A heavier plane needs more airspeed to produce enough lift to take off or not stall.

To explain why these speeds change with weight:

V_A design maneuvering speed: this is because it takes a lot more force to move a heavy plane than it does a lighter plane. On a lighter plane, the surfaces deflect, apply a force, and the plane moves. Once the plane is moving, the surfaces move too, which lessens the force on them. A heavier plane will move less, which means the surfaces are applying more force and for longer to move the plane the same amount.

V_X best angle of climb speed
V_Y best rate of climb speed: A heavier plane will need to produce more lift, and therefore have more drag, which affects climb performance.

V_R rotation speed
V_S0 landing configuration stall speed
V_S1 "specific" configuration stall speed: A heavier plane needs more airspeed to produce enough lift to take off or not stall. Here is why:

Among other things, lift is dependent on speed and angle of attack. So at a given speed, angle of attack can increase to generate more lift. The plane will have a maximum angle of attack above which it will stall. Since these are minimum speeds, they are at the maximum angle of attack, so speed must increase to provide the lift.