Does "anti-constitutional" mean against the German constitution specifically, or the concept of constitutions?

Specifically the German constitution. Or as also worded in the law "the free democratic basic order of the FRG" -"die freiheitlich demokratische Grundordnung der BRD".

What this phrase means specifically is defined by decisions of the federal constitutional court and includes things like basic human rights, checks and balances, the independence of courts, the multi party system etc.

Disrupting or trying to abolish those basic democratic laws is considered as trying to build a dictatorship or other form of unjust system.

I don't know the specifics about the KPD case but there are German communist parties, for example the DKP. It's just that the KPD is considered undemocratic.

In addition to what others have commented, communist and all other flavors of political parties are protected under the german constitution as long as they aren't anti-democratic or call for violations of basic human rights. That's because the right to form a party and express your political opinion is also protected in the constitution. So ironically it is really hard to ban fascist parties because the highest court would have to prove that their exercising their freedom to form a political party is in conflict with other basic rights and freedoms.

Communism?

Anti-constitutional here means directed against the Basic Law of the FRG or the constitution of Brandenburg (federal state).

The Basic Law does not explicitly ban socialism, AFAIK.

The ban on Nazi imagery is kind of necessary for a state patched together in the post-liberation Allied occupation.