The image is real, and the clouds are moving, they just move slow, as clouds do.
The unnatural appearance arise because they are taken by DSCOVR at an extraordinarily large distance, from a place called L1, more than one million kilometers away!!!.
The rotation of the Earth in the animation makes it difficult to distinguish the movement of the clouds, but if you just track one single area as shown in the animation below, the movements of the clouds becomes evident:
The zoom area shown in the right was extracted from the circled area in the left.
For the above animation I downloaded the original frames at high resolution from here. The resolution was reduced so that it is smaller than the limit of SE (2MB), but you can see it at full resolution here.
Here you can see a closeup on the cropped section:
The changes and movements of the clouds are quite obvious now. Aren't they?
The overall deformation is mostly due to perspective changes as Earth's rotates, but it is clear how some clouds change, and some dissolve completely during the video, like the elongated one in the upper left corner of the cloudless patch shown in the crop. If you put attention the vanishing of that cloud can be seen in the original video.
Another effect that might make the image looks weird, is the fact that the images used for the red, green and blue channels of the image are taken 30 seconds apart, and due to the displacement of the Earth and specially the Moon during that period, it creates an edge effect that is very clear in the full resolution image and might contribute to the fact that the moon seem to be pasted in the image, and the overall weirdness of the image. Here a close-up to the Moon showing very clearly the edge effect: