They took everyone, but not everyone was supported financially by the government. It was kind of like the procedure when claiming asylum nowadays.

It was regulated in the "Bundesnotaufnahmegesetz" http://www.bgbl.de/banzxaver/bgbl/start.xav?start=%2F%2F*%5B%40attr_id%3D%27bgbl152032.pdf%27%5D#bgbl%2F%2F*%5B%40attr_id%3D'bgbl152032.pdf'%5D__1413955689701

West Germany held that there was only one Germany and therefore all East Germans were West Germans by default. Today, South Korea has a similar policy with regards to North Korean refugees.