The French Revolution was a very, very long a complex event. Your question conflates two different things. There were many points during the revolution when riots and/or mob actions helped shape events, and sometimes those mobs killed people. But the Reign of Terror, the systemic execution of purported enemies of the Revolution (often via guillotine) was not a mob action. It was the organised, official, formal policy of the elected government of The First Republic.

To a certain extent, at least one mass uprising (the September Massacres) and the beginning of the Reign of Terror do share a cause: in September 1792, France was at war with a coalition of other European powers. The war was going badly for France at that time and the coalition was closing in on Paris. Many believed that part of the reason the war was going badly was that senior military officers were sabotaging the war effort. (Under the monarchy, almost all senior officers were aristocrats.)

As the coalition approached the capitol, the government called on citizens to march out and defend it. Fears that, with most able-bodied ardent revolutionaries out of the city, counter-revolutionary forces might try to take over helped spark the massacres, with mobs bursting into prisons to kill prisoners and lynching suspected counter-revolutionaries in the streets.

Despite the riot, the citizen-led defence of the city actually worked, and after a victory at Valley, France actually started wining the war. However, the fear that counter-revolutionary forces inside the country were plotting to overthrow the government never went away, eventually leading to an official policy of state-sponsored terrorism meant to crush any possible opposition to the revolution. Even so, the majority of the victims of the terror were not aristocrats --- politicians, priests, military leaders and ordinary people suspected of anti-revolutionary sentiments made up the majority of victims.