In short: Because this was a legally-proscribed method of execution. Guillotine was adopted/built to be a quick, equalizing and humane method of execution. It is actually a very quick method, and it uses one machine as the opposite of many weapons/people needed for a firing squad. All of this may sound horrible to us, but it makes sense in the context.

It is important to understand the context in which the guillotine was developed. Before the Revolution, the Ancient Regime practiced many different execution methods (beheading, hanging, burning at the stake, breaking at the wheel...) The punishment depended on the crime committed and on the person's background. Decapitation was generally the right of noble people - it was considered the quickest and least humiliating way to go. Compared to other methods, well, you can see why it was seen like that.

The Revolution changed this. Torture was banned, and as a principle of equality, it was adopted that all the condemned would be executed using the same method. No more privileges for those of "higher" birth. Instead, everyone would receive the same execution method - beheading. What was once a privilege of nobility became the right of all.

But this posed some technical problems. Nobles were a minority in France; if all death penalties are to be done using beheading, this would mean more, eh, tools. Charles Henri Sanson, the royal executioner, pointed out potential difficulties: swords had to be sharpened after each use, which meant getting more swords (and executioners had to pay for that themselves). Another issue - which proved to be crucial - is that there were numerous problems about beheading with a sword. It required the victim to be very still (and mentally composed) in order for the executioner to perform the beheading quickly and in one strike.

We might think: "well, you never saw that as a problem before", but remember, only nobles received that punishment before, AND nobody cared about being humane or quick. It was believed that the common folk might lack composure or knowledge on how to hold still. The whole thing seemed too complicated and impractical. There was a fear that it would not be possible to achieve a quick and humane death with this method.

Then the solution came - use a machine. A contraption that would immobilize the victim so they cannot move (for their own good), thus providing a quick, humane death, in line with the new sentiments about equality and lack of torture. (Again, we might not see it like that from our POV, but remember that they were coming from the experience of pre-execution torture and lengthy executions such as breaking on the wheel. Also, the beheading/guillotine came as a compromise - some people, including Joseph-Ignace Guillotin, were for the abolishment of the death penalty altogether. But many disagreed, so all that could be done was to push for a method that, abolitionists hoped, would ensure quick death without much suffering.)

The guillotine, as built in 1792, was not the first contraption for beheading (it was inspired by decapitation contraptions that exist(ed) in the past in various countries), but this one was built around the abovementioned principles (quick death, no torture, effective without the need of special skills from the part of the executioner or the victim).

The execution with a guillotine is relatively quick, all things considered. It is said that during the Terror, the average execution took a little more than a minute.

Now, at the time the guillotine was proposed (1791), nobody knew it would be used so much in the following years. (There are different discussions on the role this quick beheading contributed to the number of executions during the Terror, but it was not built for the Terror, because it was before the Terror).

Speaking of which, there were executions by the firing squad during the Revolution. These were applied in the context of the military during the war (such as the one with Austria and Prussia) that was ongoing during the Revolution.