Equation of the Center constant factor
I don't have a complete answer, but I do know where the $7.6$ factor comes from: It equals
$$(2×0.0167)×(24×60)/(2 \pi).$$

$0.0167$ comes from the eccentricity of the Earth, and the factor of $2$ comes from the above equation. $24×60$ converts from days to minutes, and $2 \pi$ is the number of radians in a complete revolution.

What I still don't completely understand is why these coordinate conversions need to be done, specifically the division by $2 \pi$.

Thanks everyone for your help. 

Edit: I now have a somewhat better explanation for the computation in this answer. 2ε sin(M) is the number of radians that the mean Sun is behind the apparent Sun on Noon on a day with mean anomaly M. Dividing the 2ε factor by 2πgives the fraction of the rotation of the Earth (i.e. one day). When the result is multiplied by the number of minutes in a sidereal day (23×60+54) yields the desired coefficient 7.63345.