The crisis was mainly within the army and the Pied-Noirs (French colonists in Algeria, roughly 1 million peole). Both felt betrayed by the withdrawal, for obvious reasons.

This dated back to 1958, when the last (civilian, roughly center-left) government of the Fourth Republic decided that maintaining the Algerian colony was not worth sending the (conscript) army there any longer. Since the army was already deployed in Algiers, it developed into a quasi-coup d'État; a few officers seized power in Algiers and Corsica (thereby directly threatening Paris). Since, as you said, most French people just wanted out of Algeria by then, the left-wing (PCF and unions) suggested distributing weapons to the workers as a measure to resist the military coup; but the government preferred to prevent the situation from degenerating into a full civil war (the scenario was extremely reminiscent of Spain 1936...).

By then, De Gaulle had been out of power for 12 years, but was (personnally) very popular in France. He also promised the military that he would remain in Algeria (De Gaulle was not a very surrender-ish name to them); there is also some evidence that at least some part of the military insurrection was supported by his party members as a way to force his recall, though there is no such evidence involving him personnally. When he was invoked in 1958 as the “providential man” preventing both the coup and the civil war, the military entered the 5th republic under the impression that De Gaulle was their obligé and would support French Algeria. When he did leave Algeria in 1962, the more extremist of the officers felt betrayed and resumed their plans for a coup; however, such a coup was now impossible, because much of the military were extremely loyal to the man who had saved France twenty years before. So they needed to remove him personnally first...