How might I attach pieces of lumber between two garage ceiling joists for the larger goal of mounting a punching bag to the lumber?
You'd have to install some more supports to your framing in order to prevent damage to the existing framing members. I'd also suggest mounting the Heavy Bag as close to a wall as possible rather than mid-span.
The ceiling joists will be carrying most of the weight and jolts. The ideal support would include a vertical supporting member under each joist and between the heavy bag. But this may not be convenient or appropriate. The (3) 2x6 boards in you picture should be installed to the joists 'on edge' and not on their 'face' (flat). They'll be able to hold greater stresses and weight.
The instructions don't mention this but it is shown in the web sites picture which shows it mounted to 2x10 joists.
I would think at a minimum you'd should sister in (2) new ceiling joists, one to each of the existing joists; spanning wall to wall. Than bridge between them with a doubled cross piece spaced for the size of your spider hanger. Also some single pieces of bridging spaced at appropriate intervals would be wise. Use 3x9 inch deck screws to join all lumber.