Why did I have to wave my hand in front of my ID card?

Movement that blocks the view of the item under inspection helps to defeat someone trying to use an overlay image on the video as a replacement for the actual item.

For instance, I could take a short video of your ID (that shows the security features) and overlay that on the live video instead of my actual ID. But by waving my hand in front, then the remote viewer can see that it is not a video overlay.

A real threat? Yes. Just look at the fake videos that we have seen where someone can make it look like someone is saying something that they never did. The technology exists and is in use.

A credible threat? Questionable, but the mitigation is no cost, easy for all involved, and simple. So, the cost of mitigation is negligible.

That means that it is not "security theatre". It actually treats a risk. But I might agree that at this point in time, it might be borderline. Next year, I might have to edit this answer.

Given that this identification was likely performed according to German law, this request was to conform with BaFin Circular 3/2017 which demands (in their non-binding English translation):

Any substitution/manipulation of parts or elements of the identity document must be countered by suitable measures. To this end, the person to be identified must be asked, for example, to place a finger over security-relevant parts of the identity document (variable and determined at random by the system) and move one hand across their face. Using stills from these movements that are cut out and enlarged, the employee must verify that the identity document, along with all the security features visually identifiable in white light, is completely covered at the right point and that no artefacts indicating manipulation are evident at the transition points.

So the stated reason for that is to uncover potential manipulation in the video feed you send them. There have to be enough and unpredictable tasks which you may be asked to make it harder for you to have a suitable substitution prepared.

Lethalcoder and others have made the point that duping the hand wave is easy to do. But that's missing the point of the request - it is an unexpected request that probably wouldn't be duped ahead of time. Tomorrow, they might ask you to show your cellphone's time, or today's paper (as if anyone reads those), or any other random item in front of the ID. This only becomes security theatre if they always ask for the same task, at about the same time in the ID process.

As to why you need to wave your hand, Schroeder explained it very well in their answer:

"Movement that blocks the view of the item under inspection helps to defeat someone trying to use an overlay image on the video as a replacement for the actual item. For instance, I could take a short video of your ID (that shows the security features) and overlay that on the live video instead of my actual ID. But by waving my hand in front, then the remote viewer can see that it is not a video overlay."