I bought one of these but it has a 10 pin connector.
How do I marry that up with the 8 pins of the ATTiny 85, or do I have to just abandon that and link pins with jumper cables and a breadboard?
Try to find a datasheet for this programmer, it should indicate the pinout of the connector.
In any case, you just don't plug this connector or anything else directly on the DIL8 Attiny85... So yes, use a breadboard, or if you can, the best is to make a small board where you can just plug the Attiny85 in a socket to program it.