The word "demagogues" gives us the clue: the frogs stand for some intellectuals, and the king wants to appear liberal by allowing them to discuss freely. Think of the Arab Spring. However, when they really start making use of their freedom, the king feels threatened and denounces as demagogues the very same intellectuals he had invited before. Think of Erdoğan and his "terrorists".

The author claims to be unaffected, which could mean apolitical - were it not for the first four verses. Rather, he recommends to keep up the camouflage even in times when free speech seems to be graciously tolerated, but may be punished later.

King Frederick William III of Prussia makes a good candidate; he needed the intellectuals in order to unite the rest of Europe against Napoleon, but this done, he feared for his own power. Let us see what happens to Erdoğan tomorrow. -

The lyrics are devised specifically to be composed for a four-part male voice choir, probably of students; the third verse must be a kind of fugue or similar. The poem has sometimes been falsely attributed to Goethe, if Google serves me.

What my dictionary says:
"Den Mund voll nehmen": to brag, to talk big
"Euch soll ...!" - presumably "... der Teufel holen!"; may the devil take you; go to hell.