How did Ole Christensen Romer measure the speed of light? How did Ole Christensen Rømer, who in 1676, shortly after Galileo's death, was said to have measured the speed of light as 220,000 km/s by timing the orbits of Io around Jupiter? 

My only guess is the moon Io orbiting Jupiter is simply a clock. It pulses away with an orbital period of 42 hours and provides to the earth, 21 hours of light each orbit; this does not change. Now, as viewed from "above" we go counterclockwise around the sun, and as we move towards Jupiter, the duration of light from Io would be less than 21 hours, whereas if we are moving away from Jupiter, the duration of light from Io would be more than 21 hours. Suppose we measure the time from Io's first appearance from behind Jupiter to its reappearance and call that its observed orbital frequency $f$ as opposed to its actual orbital frequency ${f_o}$ which is 1/42 hours.

He must have figured out the Doppler effect and this equation...
$$f = \left( {1 + \frac{{\Delta V}}{C}} \right){f_o}
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I hashed through the numbers myself and I got an answer of about 10 minutes more or less from 42 hours depending on which way the earth is heading with reference to Jupiter.
Ole Rømer did not measure a change in the frequency of light. He measured an apparent change in the orbital period of Io, one of Jupiter's moons.

The orbit of Io can be measured very accurately by observing when it enters or leaves the shadow of Jupiter. When the Earth is moving away from Jupiter, Rømer noted that the orbit of Io appeared to be very slightly longer than when the Earth was moving parallel to Jupiter. 

Suppose the orbital period of Io is $p$ seconds. If you are not moving and you observe an eclipse at time $t$, the next eclipse will be at time $t+p$. However, if you are moving away from Io, the next eclipse will be seen at $t+p+x$, where $x$ is the time that light takes to travel from your position at time $t$ to your position $p$ seconds later.

The delay over a single orbit was too small for him to measure (about 30 seconds). But the delay was cumulative, and over about 30 orbits of Io, the orbit was delayed by about a quarter of an hour. He noted:


  [This] appears to be due to light taking some time to reach us from the satellite; light seems to take about ten to eleven minutes [to cross] a distance equal to the half-diameter of the terrestrial orbit. 


Of course, Rømer didn't know the actual size of the half-diameter of the terrestrial orbit (the astronomical unit) However putting the modern value of about 150 million km this gives a value of the speed of light at about 220000km/s, which is not so far from the modern value of about 300000km/s