But how do we know that this is the speed of light? Why wouldn't it be higher or lower?
Römer's observations of Jupiter's satellites. Source: wikipedia.org
The speed of light was first measured by the Danish astronomer Olaf Römer. In 1666-1668 he made observations of the satellites of Jupiter and noticed a pattern that could only be explained by the finiteness of the speed of light, and at the time the speed of light was thought to be infinitely high. A little later, in 1672-1675, Römer, working in France with Cassini, noticed that when the Earth and Jupiter get closer, the time between eclipses of Jupiter's satellites decreases, while when the planets move away, on the contrary, it increases.
Based on this, Römer and Cassini calculated the speed of light and obtained values of 135,000 km/s and 220,000 km/s, respectively. Despite the substantial error, the result is striking, because at that time astronomers had rather primitive instruments and, accordingly, data on distances between objects. (This is why there is such a big difference between Römer's and Cassini's calculations.)
Since then, the speed of light has been refined many times due to improvements in measuring instruments and, according to modern data, it is 299792.458 km/s.
Separation of light in a prism. Source: wikipedia.org
In the 19th century, James Maxwell managed to prove that light is an electromagnetic wave. He derived a system of equations that describes the behavior of stationary charges, charges in motion, and electric currents, and combines electrodynamics and magnetism. These equations are now called Maxwell's equations.
Maxwell's equations carved on the foot of the James Maxwell monument. Source: wikipedia.org
The system has several possible solutions and one of them is oscillating, phase-matched and perpendicular to each other electric and magnetic fields - electromagnetic waves. Maxwell calculated the speed of their propagation and obtained approximately 300,000 km/s, i.e. the then known speed of light!
Fluctuating, mutually perpendicular, phase-aligned electromagnetic fields. Source: wikipedia.org
This speed has since popped up in all sorts of fields of physics:
This is the speed at which any massless particles move
It is the speed at which electromagnetic oscillations propagate
This is the speed at which gravitational waves propagate.
It's the maximum possible speed at which information can be transmitted.
It connects mass and energy (the famous E = mc²).
But why is it the speed of light?
The speed of light is one of the fundamental constants of our universe, from which many other constants are derived. If the speed of light were faster or slower, atoms might not have formed, which means galaxies, stars, planets, and, eventually, intelligent life that can conceive of the speed of light would not have appeared.
Proponents of creationism see this as proof of intelligent design, that a higher intelligence assigned to the fundamental constants exactly the values necessary for the existence of the universe and the emergence of intelligent life. But creationism is not the only possible, much less the most likely, explanation for why the constants have the meanings we know they do.
Most scientists adhere to the anthropic principle, which says that we see the universe as it is because it is this coincidence of constants that gave rise to intelligent life, and if their values were different and we simply would not exist and no one would think about these questions.
According to the multiverse theory: every time some universe is born, it acquires some set of fundamental constants and depending on what values they acquire, the universe may immediately collapse, explode or continue living in some form and there simply may not be anyone there to ask such questions.