Lennart Regebro's answer somewhat correct, but very generalized, and fails to address the core issues that were at play:
It must be emphasized that Robespierre's fall from power was not due to anti-revolutionary forces casting him down. On the contrary, Robespierre was deposed by his fellow revolutionaries, who believed that he was about to engage in a coup d'état that would have been decidedly anti-revolutionary. Robespierre was no leader of the revolution, and when most of the important action was going on, Robespierre was actually hiding out! His rise to great power came only after much of the dirty work had already been done - he was an opportunist who seized the moment only after others had done "the heavy lifting."
That being said, the real story of Robespierre's downfall and the true answer to this important question, one of the most perplexing in History to those not familiar with the details of that most terrible period, The Reign of Terror, requires a bit more discussion:
At the time when Robespierre fell from power, the situation in France was somewhat akin to what we find later on in the USSR and other communist states, which had an official state government that was controlled by a political party, and acted as a rubber stamp for that party. The Soviet government was not the Communist party - but the Communist Party's leader's were also the leaders of its government and the big decisions were made by the Party and then authorized by the government and promulgated as Law.
Similarly in France, the official state government consisted of the National Convention and the smaller, more powerful Committee of Public Safety. Robespierre was the leading member of both of these bodies.
However, Robespierre's true seat of Power was in the Jacobin House of Lords: The Party of Robespierre - the bastion of Sansculottism: Les Misérables, the foot soldiers of the Revolution from whence Robespierre's enormous political power ultimately derived. The Jacobin House of Lords represented the most extreme faction of the Revolutionaries and was Robespierre's private echo chamber - somewhat analogous perhaps to the Communist Party organs that allowed the rise of men such as Stalin and Brezhnev. The government as a rule was submissive to Robespierre and his party. But not always....
The government embodied in the the National Convention, although overwhelmingly Jacobin and revolutionary, was comprised of many more moderate factions, in particular the Dantonists, led by Georges Jacques Danton. After the execution of Louis XVI by the Revolutionaries, and the elimination of the Girondists, who advocated a moderate Constitutional Monarchy along the lines of Great Britain, a schism developed between the most extreme Jacobins led by Robespierre, and the Dantonists, who were also Jacobins, but had grown weary of the Reign of Terror and advocated leniency for some of the aristocrats and a return to some sort of normalized society.
Robespierre's first great mistake was allowing Danton to be Guillotined by a "Kangaroo Court". Before meeting his fate, he addressed the crowd "My only regret is that I am going before that rat Robespierre." Danton's demise sent shock waves through the Convention, as Danton was a hero and a leader, and others who were not part of Robespierre's cabal began to fear for their own lives. The idea began to circulate that Robespierre wanted to establish a dictatorship with the support of the Jacobin House of Lords, and eliminate entirely the Convention as the working body of the French Revolutionary government.
For his part, Robespierre had become increasingly isolated from the workings of government, devoted his attention mostly to the Jacobin House of Lords, and was developing a private personality cult bordering on a quasi religion, contributing further to the reservations and suspicions about him.
Eventually a rumor surfaced that had some reasonable credibility: Robespierre was planning to storm the Convention Hall with troops and send them all to the guillotine - in a word, a coup d'état, that would make Robespierre a dictator and his cadre of supporters in The Jacobin House of Lords the facilitators of his rule: This coup d'état, had it succeeded, would have represented a great loss to the Revolutionary cause, setting the stage for further abuse and tyranny (albeit not by the Ancien Régime ) which it was goal of the Revolution to prevent forevermore. Robespierre's endless ambition showed him to be an enemy of the Republic. In a preemptive move, the Dantonists and others in the Convention had Robespierre arrested, and in spite of some machinations on Robespierre's part to escape, the tide had turned against him in the National Convention - thus he was subdued.
It is also important to note that although Robespierre is generally identified with the guillotine, its use was not Robespierre's innovation. It was the tool of the Revolutionary Government at large - the most expedient way of disposing of its enemies. Subsequent to Robespierre's demise, the use of guillotine continued, particularly to rid the Revolutionary Government of those extremists who had supported Robespierre.
Source: The French Revolution, a History by Thomas Carlyle; Particularly Chapters 3.6.VI, VII.
