How does the dihedral angle work?
Basically, dihedral effect is that during banking, the 'lower' wing will experience a higher angle of attack compared to the 'higher' wing, and a result, a greater lift. The resulting net force and moment reduces the banking angle, reducing stability.
Consider a wing with a dihedral angle $\Gamma$ with a forward airspeed of $u$. If the sideslip angle is $\beta$, the wind due to sideslip is $u \cdot \sin\beta$. From geometry, the normal velocity induced due to dihedral, $v_{n}$ becomes $u \cdot \sin\beta \cdot \sin\Gamma$.

Image from Stability and Control of Aerospace Vehicles
Note: The notations are different in the figure; but the principle is the same.
For our purposes, we can take the sideslip velocity ($u \cdot \sin\beta$) as $v_{y}$. Now, consider two sections from the wing - one each from the 'lower' and 'higher' sides. The induced velocity is of the same magnitude in both the sides, while the direction differs, as can be seen from the above figure.

Image from people.rit.edu

Image from Stability and Control of Aerospace Vehicles
For small angles, $v_{y}$ is nearly equal to $u \beta$. The induced angle can be given as,
$\Delta \alpha = \frac{v_{n}}{u}$.
From the earlier relations, we have,
$\Delta \alpha_{1} = \beta \cdot \sin\Gamma$, and $\Delta \alpha_{2} = -\beta \cdot \sin\Gamma$.
Because of these induced angles, the lift on the downgoing wing increases by $\Delta L$, while of the other one decreases by $\Delta L$. The net result is that the 'lower' wing experiences an increases lift, causing a rolling moment, which causes the banking angle to reduce.

Image from Stability and Control of Aerospace Vehicles