I have a hard time understanding how the world lines of tachyons in a space-time diagram imply the violation of causality. Can you explain why that is?
Basically, because events on a tachyon worldline are not well-ordered in time for all normal (i.e., non-tachyonic) observers.
To normal observers, any two events on a tachyon worldline have a spacelike separation. It can easily be shown (using Lorentz transformations) that if events A and B are separated by a spacelike interval, then there are 3 classes of normal inertial reference frame. In some frames, A & B are simultaneous, in some frames, A happens before B, and for other frames, B happens before A.
This relativity of simultaneity is illustrated in this spacetime diagram from Wikipedia.
So if information is transmitted tachyonically from A to B, it's possible to use sublight speed information transmission to send information back from B to A.