Wikipedia has the answer, but it requires some assembly
Wikipedia: Leopold III contains the phrase,
Leopold's controversial actions during the Second World War resulted in a political crisis known as the Royal Question.
Royal Question is hyperlinked to another Wikipedia page, which explains,
The "Question" at stake surrounded whether King Leopold III could return to the country and resume his royal duties as King of the Belgians amid allegations that his actions during World War II had gone contrary to the provisions of the Belgian Constitution.
. . . .
The crisis emerged from the division between Leopold and his government, led by Hubert Pierlot, during the German invasion of 1940. Leopold, who was suspected of authoritarian sympathies, had taken over command of the Belgian Army at the outbreak of war. Considering his constitutional position as Commander-in-Chief to take precedence over his civil role as head of state, he refused to leave his army and join the Belgian government in exile in France.
So in short, Leopold felt that his obligations as Commander-in-Chief took precedence over his obligations as head of state. The Commander-in-Chief had to remain with the military; the head of state would have been obliged to follow the Belgian government into exile.
I am not a scholar of the Belgian constitution, but to my mind, the critical phrase is that the King "refused to obey the government". In a constitutional monarchy, this is very likely to generate a constitutional crisis; the protests you mention are the consequence of that crisis.
I just checked the Wikipedia page on the Belgian Constitution
The Constitution of Belgium (Dutch: Belgische Grondwet, French: Constitution belge) dates back to 1831. Since then Belgium has been a parliamentary monarchy that applies the principles of ministerial responsibility for the government policy and the Trias Politica.
Interpretation of the law requires a subtler mind than mine, but this confirms my hypothesis that Belgium is a parliamentary monarchy, which means that the monarch cannot/should not defy the parliament. The precise division of executive power seems to be governed by Articles 34 through 37, but if the King is subject to a foreign power while the government is in exile, that will introduce constitutional questions that make me glad I'm not a lawyer.