The voiceover does not exist in a vacuum, like example sentences in a textbook; you have to watch the video and listen to the actors, too. The opening shot demonstrates that the narrative "now" is a point well into the meeting: Saint-Just scrawls his signature on a document and lays it before Robespierre. Simultaineously, the voiceover gives a date and hour for this Reference Time ...
In the early hours of July the 28th, 1794 ...
... and continues with the background to this moment. This is the paradigmatic use of the perfect: it names a state current at RT which arises out of eventualities before RT.
... the leaders of the French Republic's Committee of Public Safety had gathered for one last time, their once colossal power evaporating with every minute. For 12 months Maximilien Robespierre had ruled revolutionary France in the name of the people.
When that is established, the narrative shifts into ordinary pasts:
But now in the name of the people soldiers were on their way to seize him. His dictatorship was over, and he was about to become the final victim of his own bloody Reign of Terror.
You may (if you have the stamina) read more about this perfect double time reference and how it anchors prior events in a current context at What is the perfect, and how should I use it?, especially §4. When and how should I use the perfect?.