In my opinion, the 9th of Thermidor isn't all that special, it was one in a series of purges of factions in the national convention and the Paris commune. The first of these was the Girondists in 1793, then the Hébertists in the spring of 1794, closely followed by the Dantonists and then the Robespierrists in the summer. Basically, differences of opinions and allegiances lead to escalating conflict and eventual mass execution because the political climate was absolutely batshit insane.

In the case of the Robespierrist purge, the conflict had a lot to do with the Terror. Other members of the Convention and the Committee of Public Safety had done things he didn't agree with, like the way the repression in Lyon was conducted by Collot d'Herbois (member of the CPS) and Fouché (of the Convention). They were backed up by others like Billaud-Varennes (also CPS, allied with Collot). This conflict dates back to the Dantonist purge, where the Dantonists gave their support to women from Lyon petitioning for the freedom of their imprisoned husbands. Robespierre suggested a committee of justice be appointed to sort the mess out and possibly let them go, but Billaud-Varennes put a stop to it to protect his allies who had been the ones who caused the imprisonments. Things in the CPS were already tense due to its nature as something of a coalition government, and this doesn't make things any less awkward.

Robespierre doesn't let the Lyon thing go, even though he drops his support for the Dantonists and helps the rest of the government arrest them. He also goes after others who have done fucked up things in the provinces, like Carrier, who drowned thousands by sinking barges in a river. While Robespierre is doing this he unofficially abandons the CPS, letting his enemies rule the government alone. Bad move. In his absence they gang up on him, but eventually he decides to make a move against them. He holds a speech in the Convention, where he makes the fatal mistake of denouncing people without actually mentioning names. That's a really dumb thing to do. In practice, Robespierre doesn't have the kind of power the Thermidoreans would later claim he did. Like with the committee of justice, his demands can be easily shot down by the CPS, but the Convention is still afraid of him. So, when Joseph Fouché goes around telling everyone that Robespierre is after them so they need to kill him, they believe him. And when Robespierre makes vague speech about how certain corrupt individuals need to be removed, he terrifies a lot of people, even if the only people he was actually planning on having killed where mass murderers.

Robespierre also had a lot of enemies in the Convention for other reason, just like any politician, and they took the chance to get rid of him in order to get rid of the reforms made during his time in government.

On the 9th of Thermidor, the day after his speech, Robespierre and his ally Saint-Just go to the Convention where Saint-Just is planning to make a speech. He doesn't get to, though. Instead, Billaud-Varennes reads an accusation against Robespierre, accusing him of abandoning the CPS, protecting the Dantonists, and aspiring to dictatorial powers. The Convention shout Robespierre down whenever he tries to defend himself, a strategy that's clearly been planned in advance. They vote to have him arrested, along with his allies in the Convention. The Paris commune rebels, and tries to defend him, but it's no use. Twenty-six go to the scaffold on the 10th of Thermidor, and another seventy follow them the next day. It's the biggest purge in the history of the Revolution, but on the 11th of Thermidor that's all it is.

It takes a while for the actual end of the Terror to happen, though. The Terror is usually said to end with Robespierre's death, but the prairial law and other terrorist measures stay for a while longer. Not to mention that the biggest mass execution in Paris happens on the 12th.

Many of the Thermidoreans were themselves bona fide terrorists and had no thought of ending it, but with the leaders of the left gone the moderates take power and end both the terror and the social and democratic reforms made during the Montagnard dictatorship. For example, the prosecutor of the revolutionary tribunal is confirmed on the 28th of July and arrested three days later. During that time, the right and opportunist former terrorists managed to turn the convention in the direction of ending the terror.

When it comes to blame, I would say Robespierre takes some of it for his poor strategy in leaving the CPS and then making vague threats. The CPS had a lot of the responsibility, but they did it on purpose. The Jacobins supported Robespierre, but being on the losing side of a factional dispute meant certain death, and many of them weren't too excited about that possibility. They chickened out, basically, though considering the Convention had control of the army they might not have been able to do much anyway.