The "solo" button will send THAT channel to the headphone jack, so you can listen to just 1 thing at a time. This is useful when trying to isolate a problem, such as feedback or distortion.
The "insert" jacks are stereo (R-T-S) and with the proper cable, you can directlt patch and outboard processor or effect into THAT channel only. This is most often used for compressors or dedicated effects.
I had one of those Fender boards, and it sure was loaded with features. The size/weight ratio left me in the cold, but for a band or a permanent set-up ..... it's hard to beat.