In many books one finds different explanations. Specifically popular seems to be that he "argued against the Copenhagen interpretation". But what did he really intend to communicate?
I for myself have tried to come up with an answer to this question. Please feel free to present the correct answer and / or critique my version:
In my point of view, Schroedinger tried to discuss the current situation in quantum physics (at that time) which seems to be somewhat equivalent to what we now call the Copenhagen interpretation (?).
He discussed how one could understand the "Verwaschenheit" or "Unschärfe" (the fuzziness/blurriness) of quantum physics, which today might be called the probabilistic character of quantum mechanics (?).
He although seemed not to be as worried about the probabilistic features itself, but more how one could use a realistic interpretation on this (since he and Einstein seemed to have a realistic point of view when it comes to the interpretation). In his famous paper from 1935 (where he introduced the cat Gedankenexperiment) he claimed that it was impossible to interpret this blurriness as just "having not enough information" (e.g. a Gibbs Ensemble) and also argued that viewing this blurriness as a objective property of quantum systems is inadequate with a realistic point of view. This (latter specific point) was made clear by the cat Gedankenexperiment as the blurriness as an objective property would lead to something that doesn't match out intuition in the macroscopic world.
So in the end, he tried to make clear, that a realistic interpretation didn't work for any of the 2 possible ways (Gibbs Ensemble vs. objective property) the current situation of quantum mechanics (in other words the Copenhagen interpretation) was able to provide. The thought experiment was used to "cross one of the 2 possible ways out of the equation".