Why can different observers agree on their relative velocity

Relativity doesn't say that different observers must measure different values for everything. Relative velocity between two observers is symmetric: if you are moving at a certain speed relative to me, I must be moving at the same speed relative to you.

(Note that I said "speed" here; actually, your velocity relative to me will have the opposite sign from my velocity relative to you, because velocity includes direction, and the direction of your motion relative to me is opposite from the direction of my motion relative to you. Perhaps that is what was confusing you?)

Can you give a reference? We can't really help you understand what the derivation you're reading is saying if we can't see the context.

 

Lorentz transformations act on coordinates, yes, but you can't just assume that that changes the magnitude of all expressions involving those coordinates. To see if relative speeds change when you change frames, you have to transform the coordinates specifying the worldlines you are interested in, and then derive the speeds in the new frame from the transformed expressions for the worldline coordinates.

So, for example, in the unprimed frame the primed observer is moving along the worldline ##x = u t##, by definition. That means ##dx / dt = u## for that worldline, whereas for the unprimed observer in this frame, he is at ##x = 0## for all ##t##.

Now transform those two worldlines to the primed frame. The worldline ##x = u t## for the primed observer becomes the worldline ##x' = 0## for all ##t##, by definition of the primed frame. We can show this by using the inverse transformation ##x = G ( x' + u t' )## and ##t = G ( t' + u x' )##; we substitute into ##x = u t## to obtain ##G ( x' + u t' ) = u G ( t' + u x' )##, which simplifies to ##x' + u t' = u t' + u^2 x'##, or ##(1 - u^2) x' = 0##, or ##x' = 0##, as desired.

Now do the same substitution for the worldline ##x = 0##. What do you obtain? And what does it give for ##dx' / dt'## for that worldline?

 

A worldline is a curve in spacetime that describes a particular object--where it is in space at every instant of time. The description of a worldline in a specific coordinate chart will be a function; in the case we're discussing, the function we're using is the function ##x(t)## giving the ##x## coordinate of the object for each ##t## coordinate.

As Nugatory said, it's an observational result. However, there are also good theoretical reasons for it, to do with the general properties that the Lorentz transformation must have. The key property for this discussion is that the transformation from the primed frame to the unprimed frame must be the inverse of the transformation from the unprimed frame to the primed frame. Working through the details of what that implies, IIRC, will turn out to require that ##u' = -u##, i.e., that the velocity of the unprimed observer in the primed frame must be the negative of the velocity of the primed observer in the unprimed frame.