What do we mean when we say that two light rays striking a screen are in phase with each other?

When the electric field due to one is a maximum, the electric field due to the other is also a maximum, and this relation is maintained as time passes.

Explanation:

Phase of a wave or light ray is the instantaneous situation of the cycle in which the wave is at a given time.

When two waves are in phase means that the maximum and minimum of both coincide in time. They are in the same point of their cycle at the same time. And this relationship is maintained as time passes.

The waves can also be visualized as the oscillation of an electric field. (usually plotted like a sine function).

So the fact that two waves are in phase means that the maximums of their electric field coincide in time.