In particular, why didn't messages being relayed from the future use up bandwidth and prevent new messages from being sent back?
You're right — the total number of bits that can be in transit is limited by the infrastructure. Each booster/shutter combination provides a fixed number of bits (the cameras can probably be shared between several shutters, and the mirrors can probably be shared between many cameras). The faster you can make the boosters and shutters, the more bits you get per channel, but it's still finite, and so if you want to receive more messages you may have to install new hardware.
Background (with spoilers) for anyone who hasn't read The Arrows of Time:
In the Orthogonal universe, time and space are perfectly symmetric. There's no light-speed barrier, and no distinction between space-like and time-like intervals. Light travels in all directions in 4-space, including directions that are future-like and past-like in the local frame of reference.
As a spaceship travels faster and faster relative to an observer, its frame of reference rotates so that eventually the spaceship's direction of motion is parallel (or antiparallel) to the observer's t coordinate. That is, you can travel into a observer's future or past. (How can causality possibly work in such a universe? Read the series to find out!)
A consequence of this is that it is possible to send messages from the future into the past (in your local frame of reference, of course). The technological implementation works like this: pick a star in your distant future and point a camera at it at time t=0, with an open shutter in the camera's line of sight at a distance d from the camera. (Using mirrors to extend the light path.)
At time t=d/c (where c is the speed of the slowest-moving light that the camera can detect), close (or not) the shutter. Light from the star arrives or not at t=0, thus communicating one bit. (Note that the diagram shows the light travelling from the camera to the star: in the Orthogonal physics this is the same as the light travelling from the star to the camera.)
If you can build an automatic mechanism that can retransmit the bit in less time than t, then you can use this as a repeater to send the bit backwards from arbitrarily far in the future (so long as the system continues to run reliably for the duration):