Refugees usually don't have enough money to compete on the open rental market in Berlin's central districts. They can apply for "WBS" which can get them an apartment that is subsidized by the state (for which the landlord has to apply upfront) and which tend not to be in the center anymore but in low-income areas where landlords have an incentive to enter that contract with the state - they are sure that they get the rent every month on time in contrast to fighting for months to get rid of tenants who default on their rent.
250 applications coming in over night were already common 5 years ago. I remember that in 2001 I was seeing an apartment together with about 50 other interested people - the ad had been up since the day before 3pm...
This doesn't just happen in Berlin. I've lived in several German cities, Berlin included, and it's really difficult to find an apartment in _any_ reasonably large/popular German city (e.g. Hamburg, Frankfurt, Munich, Köln).
It's a Germany-wide problem, that just happens to be most acute in Berlin. Unless you're really fast, super aggressive, or can leverage your network to talk to the people moving out, you'll have a difficult time in finding an apartment.