At a point in space you have two waves arriving with amplitudes $A_1$ and $A_2$ and with wave 2 in advance of wave 1 by a phase angle of $\delta$.
I have chosen amplitudes just to be able to differentiate between the two waves.
It is not unreasonable that you add displacements if you think of one wave trying to displace a particle of the medium through which the wave is travelling by a certain amount and the other wave trying to displace the same particle by another amount.
You add those two displacements to find the resultant displacement.
Since we need to add two sinusoidal functions with the same frequency but which differ in phase, phasor addition can be used.
Using the cosine rule the resulting amplitude $B$ is given by $B^2 = A_1^2 + A_2^2 + 2 A_1 A_2 \cos \delta$.
Since the intensity $I$ is proportional to the amplitude squared
$I \propto A_1^2 + A_2^2 + 2 A_1 A_2 \cos \delta$.
If $A_1=A_2=A$ then $I \propto 4A^2 \cos^2 \left (\frac \delta 2\right)$ and the intensity graph is shown above.
A phase $\delta = 2 \pi$ corresponds to a path difference (pd) of one wavelength etc.