The two patterns on the right show the illusion.
The two patterns on the left are in the same horizontal position and we perceive them correctly aligned. The two patterns on the right are also in the same horizontal position. However, we perceive them misaligned. This is the drifting pattern illusion.
The difference between these two situations is that the internal texture in the patterns on the right drifts. The drifting texture in each pattern is perceptually shifting the whole pattern in the direction of motion. So, the pattern drifting to the right looks displaced to the right and the pattern drifting to the left looks displaced to the left. This causes the misalignment.
According to compensation theories the illusion occurs because the brain interprets the patterns whose internal textures drift as moving objects. Moving objects might be perceived shifted in the direction of motion to compensate neural delays (see flash-lag illusion).
You can experience another related illusion: Look for 30 seconds to the white dot on the right. Then, shift your gaze to the white dot on the left. Motion adaptation causes that the stationary patterns are not perceived aligned anymore.
Position perception: Influence of motion with displacement dissociated from the influence of motion alone. Linares, D., Holcombe, A. O. Journal of Neurophysiology, 100, 2472-2476.