As commenters suggest, you should be looking into serial protocols. The ATTINYs can support pretty much any protocol, so you want to pick one that will best fit your needs.
You could use a one-way protocol by assigning an address to each node (a ATTINY with LEDs), connecting them all to a common serial bus, and then sending packets like "Hey, address #4, switch to RED" from the controller. Each node would ignore any packets not sent to its address.
If these nodes are going to be far apart, then getting the signal to travel reliably is likely going to be one of your challenges.
If you are willing to add more hardware, then RS485 (and related current loop systems) are a time proven way to reliably move data over long wires and the hardware can be very cheap. For example, these adapters cost less than $2 each.
If the nodes are not going to be too far apart and you do not want to add extra hardware, then I'd recommend using the chip's USART to read rs232 style serial data at a relatively low speed - the slower the more realizable. Connect the TX pin on the controller to a wire, and then connect that wire to the RX pins on all the nodes. You must also connect all the grounds together.
If the nodes are all going to be in a row, then you can get slightly more complicated and daisy chain the nodes so that each one's RX pin is connected to the TX pin of the next one all the way back to the controller. This can effectively extend the maximum total length the data can travel.
Of course there also also many, many other ways to do this and the best one is really the best one for your particular situation so hard to pick without knowing more details and requirements.