I will take a shot at it, starting with the intuition behind it, which is easiest to explain if we assume one large star and one small one, so that the former seems to be almost fixed, with the smaller one orbiting it.

Unless the star's orbit is perpendicular to our line of sight, the small star will alternately move away from us and towards us as it performs each orbit. If Ritz is correct, then as it moves towards us, the light from the star would be sped up by the star's velocity in our direction, and as it recedes, the light would equivalently be slowed down.

Over the time it takes for the light to reach us, the faster light from that part of the orbit when the star is approaching us would tend to catch up with the slower light from the previous part of the orbit when it is receding from us. The longer the light is in transit, the more catching up it does, to the point where it might even pass the earlier light.

This would change the relative timing of the arrival of the light on Earth from different parts of the orbit, reducing some intervals and stretching others.

On the other hand, if the light travels at constant speed, the timing is undistorted as seen from Earth (other than the small change due to the differing distance, not speed, over an orbit, which is small and does not increase with the time of flight of the light.)

This means that if we attempt to reconstruct the orbit of the star, we get different results depending on which assumption about the speed of light that we choose. For example, if it is a circular orbit and the speed of light is variable, the half-orbit when it is receding would seem longer to us than the half-orbit when it is approaching us. If, when observed from Earth, both half-orbits are the same duration, we can deduce that the speed of light is constant, as a correction for variable c would give an orbit that is not consistent with Kepler's laws of orbital motion (we can identify the start and end of the half-orbits from the rate of change of the doppler shift, which would be zero at the furthest and closest distance, independently of the speed issue.) (Note that this is a simplified example, as I imagine we would like to be able to make a determination with less than one full orbit's data, and with elliptical orbits in arbitrary orientations, and for a wide range of stellar mass ratios.)

As for the equations, 1 is just saying that the radial speed is some function of time. The specific function depends on the relative masses of the stars, the orbit's size and eccentricity, and the orientation of its plane and axes relative to Earth, but De Sitter does not need to specify what it is, as later on he merely has to show that, as observed from Earth, the motion would be measurably different depending on which assumption is made about c.

Tau is the time of arrival of light emitted at t, in the form of t + the time of flight at c + an adjustment for the radial speed of the star. That adjustment is zero if the speed of light is constant, and, to a first approximation, (u * distance) / (c*c) if c is variable.

Equations 2 give the deduction of the radial velocity from the observations made on Earth, as a function of time of observation rather than time of emission, and the key point is that they differ according to which assumption about c you use. It is significant, I think, that the correction term au, in the variable c case, varies with time, as u, the radial velocity, itself does so. This shows that the reconstructions cannot both be consistent with Kepler's laws, and the assumption that gives a consistent reconstruction is correct.

The rest of the paper establishes that this is a large enough effect to be measurable in many cases, and reports that those measurements show c to be constant.