What is the logic behind downsizing climbing shoes?

The logic is that tighter shoes gives you more precision on smaller holds, and helps keep the shoe in place when doing things such as heel hooking and general heel usage.
If your heel moves in your shoe while heel hooking, that could be an indication that your shoe is either too big, or that it doesn't match the shape of your foot. Either way, it can, at some point in your climbing journey, reduce efficiency.

Same with being precise on smaller foot holds. If your shoes are not tight enough and are loose, you may not be able to stand on certain foot holds, specifically those very small and easy-to-slip-off-of foot holds.

This is why climbers usually downsize when choosing climbing shoes.

With all that said, though, I feel it is very important to mention and to keep in mind that the level of precision you would need from the climbing shoes depends on the level of climber you are. Just because you have professional, expensive, and tiny climbing shoes does not mean you will become a professional climber. If you have just start out climbing, there is no point in downsizing. Even if you have been climbing for a year or two, downsizing may not really be required. It may not even be dependent on the amount of time you have been climbing for.
Usually, for boulders for example, routes up to v7-v8 don't really required such precision. Many times you can even see more advanced climbers climbing v7 and v8 and higher even without shoes, so even the climbing level as an indication should be taken rather lightly.

My belief is that when you will need tighter climbing shoes, you will know it, because you won't be able to climb that specific route because your shoes will just be too loose. Until then, especially if you have been climbing for under a year or two, very tight climbing shoes are really just not needed. In my opinion, of course.

Another reason for not downsizing when you are just starting out: It hurts!
Climbing shoes are not comfortable. They can be more comfortable in comparison to other climbing shoes, but as a shoe in general, they are just not comfortable, and can even hurt. Even more than that, if you try to jam your feet into small and tight climbing shoes without properly conditioning your feet to gradually get used to that kind of pressure, you could end up injuring your self in some way.
Also, as mentioned above, there will be no real benefit to you if you use professional and super tight climbing shoes as a beginner compared to using beginner climbing shoes since the routes you will be climbing probably won't include holds where you would need that level of precision, and the tight professional climbing shoes would have no added benefit in comparison to looser, cheaper, beginner level shoes.

One more thing, just to point out that some people, including myself, actually go up in size from their street shoe size. I go up about 1 size from my street shoe. Going down even half a size is just too tight, too painful, and just hurts my climbing rather than improving it.