Short answers, pulled from the story.
Ole Rømer proved that light travels at a finite speed in the year 1676. He made this discovery by observing the eclipses of Jupiter's moon Io and noting timing discrepancies caused by the time it took light to travel across Earth's orbit.
James Clerk Maxwell published his paper proposing that light is an electromagnetic wave in 1865. He derived a mathematical relationship showing that the speed of these waves depended on the electric permittivity and magnetic permeability of the vacuum.
The General Conference on Weights and Measures redefined the speed of light as a fixed constant in 1983. This decision fixed the speed of light at exactly 299,792,458 metres per second by definition and redefined the metre based on the distance light travels in vacuum.
Galileo Galilei was the first person to attempt measuring the speed of light in 1638. He tried to measure the delay between uncovering a lantern and seeing light from a distant mirror but concluded that light was too fast to measure with human senses.
Albert Einstein showed that as an object with mass approaches the speed of light, its energy requirements increase toward infinity. This makes it impossible for any object with mass to reach that speed, establishing the speed of light as the cosmic speed limit for all matter and information.