The unopportunistic and unattractive truth (I'm not afraid of saying it) is that Berlin is a red-red-green governed city and rents are one of their most important lighthouse projects. Also, the market does not, and cannot work because the market is actively prevented from working.
The market is driven by offer and demand. Demand has been going up, so consequently, either prices go up, or offer has to go up in sync with demand.
Offer, however, depends on availability, and availability is being actively sabotaged, both directly and indirectly (by the state government as well as the same city senate that complains about too few apartments being available, and too expensive).
Like in every good lie, there is a small grain of truth in here. Rents in Berlin did explode. However, they exploded from virtually nothing to well below average, which is about half of what you pay in some other places.
You get a smallish apartment for around 10€ per square meter in Berlin, and it's around 11-12€ for a mid-sized or larger one (feel free to consult e.g. immowelt which has detailled comparison charts for virtually every city).
For the smallish kind of apartment, you'd pay 12€ in Cologne and 20€ in Munich, for the mid-sized (and larger) it would be around 11€ and 18€, respectively. For Frankfurt, figures are around 14-15€, for Stuttgart around 13€.
Sure enough... you can get an apartment in a mid-sized non-spectacular city (say, Kassel, no offense intended) for 1-2€ cheaper than in Berlin, but you can hardly compare some moderately unimportant city in the middle of nowhere with living "Downtown Capital City, the most exquisit place to be". Which, as it happens, is a city roughly 12 times as large, too. I mean, seriously. These are just not the same things, you cannot compare them.
Thus, from a purely objective and non-ideologic point of view, there is actually nothing to complain about. Living in Berlin costs half as much as living in Munich. Whining about a non-issue, and being served by red-red-green in a no-effect, but very visible manner.
For red-red-green, it is political gold to prevent the evil class fiend from stealing from the poor, and capping rents is much easier to do than to actually do something about the general problem of housing being behind demand. It doesn't even matter that it makes the situation worse instead of better as there is no big incentive for house owners to invest in modernization, or for ground owners to build (if they get a permission at all!). It simply isn't worth it.
Of course the city could build affordable housing, but oh my, that would be a lot of work, and it's not nearly as effective politically than to point a finger at the class fiend.
This is not limited to Berlin, but Berlin in particular has been giving out significantly fewer permissions to build than to cover the need during the last years (with fewer every year). Which they do away with pointing at overhang and the fact that the rent cap wouldn't apply to the new apartments.
The regulations for building new houses get crazier and crazier (which means more cost-intensive) so with the rent cap in place it is almost no longer profitable to build at all -- if you get a building permit at all. The best thing that a ground owner can presently do is... do nothing. Just wait and see how prices go up.
Given the situation that is explicitly created by the government on every level (ECB negative interest, Bund, Länder, Städte, alike) it will be "fun" to see what becomes of it. Surely, the situation will not improve any time soon.
The subvention of compensation for PV is going to be cancelled this year unless Altmaier gets his stuff together before the 52GW ceiling is hit (unlikely...), which is a truly smart thing to do when you are actually shouting "Energiewende" all over, and the planned EnEV2021 which is yet crazier than the present one will make building affordable apartments for rent even less attractive than it is already. So if you will be looking for an apartment in 2022-2025, good luck.
Be that as it may, in Berlin, people currently don't have much to complain about compared to other places.