For Americans who don't have a similar system, a "government collapse" isn't as big a deal as it sounds. It sounds like there's a complete breakdown in law and order and nobody's in charge. Really what happens is that the arrangement that so-and-so will be prime minister and his cabinet will be X, Y and Z is off.
Sometimes it means there are new elections. But, sometimes (as in the French system) it just means that the various representatives all negotiate among themselves to choose a new prime minister. The President then appoints that person. It can vary from the president rubber stamping the decision, to the President getting involved in the negotiations and playing a key role in choosing the next PM. Once the President makes it official, that person becomes PM and then chooses a new cabinet. Before a new PM is chosen there's a bit of chaos. The government can still vote on things, but the normal process is disrupted because there's no "first among equals" to lead. In the case of France, normally the President doesn't (or shouldn't) deal with the day-to-day running of the government. But, during the previous government's collapse Macron stepped in to do many things the Prime Minister would normally do.
One minor twist here. In theory, a French President is supposed to handle foreign policy and defence. The Prime Minister is supposed to run domestic things, including the day-to-day government functions. One reason why this government lasted 14 hours (or 27 days if you count his full time as PM) is that Macron was seen as having too heavy a hand in picking not just the PM Lecornu (picking the PM is technically his job as President), but also in picking the PM's cabinet (which is supposed to be something the PM does himself). As soon as Lecornu announced his cabinet, the rest of the elected reps saw that it was essentially the same as the one they just voted down a month ago. They said they weren't going to work with Lecornu's government, so Lecornu quit immediately.
