I’ve read some article about the „Böhmermann case“ in english-language media.
I don’t think the article draws the whole picture. Therefore I’d like to explain the issue to you from my point of view.

Germany is a country with a difficult past. Therefore, freedom-of-speech law is a bit difficult, too.
There is, for example, a law for the prosecution of people who deny the holocaust happened.
Also, there is a law against raising hate against parts of the population („Volksverhetzung“), which is intended to e.g. ban calling for the murder of all red-haired, protestants, or dark-skinned people.
And there is a law against defamation.
But there are limits to these laws. Some things one might say are deemed too unspecific to be defamatory, like „All men are pigs“. Also, calling to kill all men, would be considered too absurd to be really offensive.
Also the context in which you say something is also considered in legal arguments. „Soldiers are murderers“ may be interpreted differently if you say it directly to servicemen, for no reason, or if you have it written on a bumper-sticker.
Actually, the german justice system tries to consider what you meant to express, it does not only consider what you actually said.

As a matter of fact, Mr. Böhmermann launched a full torrent of (bad, racist, obscene, disgusting) curses against Erdogan.

But the context is the following: After another TV station produced a (in comparison, harmless) song about president Erdogans questionable stance on freemdom of speech, democracy, checks & balances, and civil rights, president Erdogan summoned the german embassador. Which is, in the world of diplomacy, a quite high level of escalation.

Mr. Böhermanns poem now is to be understood, in this context, as making fun of Mr. Erdogan for summoning an embassador for — in comparison — no reason. The poem is super-offensive to illustrate that — if ever — this level of insult would have justified summoning the embassador, not the rather harmless song of Extra3. To make this point, the argument of the defenders of Mr. Böhmermann goes on, it was indeed required to really speak out all the slurs. Therefore the declamation of the poem had to be considered a form of legitimate satire.

Juridical and public opinion on if Mr. Böhmermann’s poem is a permitted satire, or an outrageous defamation, is split in germany, though. Quite some people, including especially germans of turkish ancient, say that no artistic or political cause could ever justify this mass of low-down offence.

Eventually the courts will have to decide if the poem was — in its context — a piece of art, or a crime.