How did Ole Christensen Romer measure the speed of light?
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