I believe the out-of-universe and in-universe reasons are identical: the actor / character on the right is much shorter than the one on the left.
First of all, the actor on the left has shoe soles that are not especially thick. Here is a close-up on his feet from the very same episode:
The actor on the right is shorter than both his compatriot and Worf. This height difference would be even greater if he were not wearing the platforms.
Out-of-universe, the Klingon pair are intended to be fearsome warriors, bred for battle. Height helps to convey this. In-universe, a short Klingon would experience a strong need to compensate.
I can find no official confirmation of this, but I believe it falls within the bounds of reason.
It is worth noting that the TNG-style Klingon costumes were designed by Robert Fletcher, who based the uniform on Japanese feudal warriors (as detailed in Reeves-Stevens, Judith and Garfield (1995), The Art of Star Trek, Pocket Books). There does not seem to be a specific inspiration for the footwear.
By the way, the boots don't have masking / duct tape. It is metal plating, as seen in the clip above but also here:
Long story made short: I believe the thickness of the platform is tailored to the actor, and is used in-universe to nurture an intimidating pose.