https://imgur.com/a/dv8ZicL

For the life of me I could not figure out how to get past this area. Jumping, grappling on the fire/blocks, and swinging didn't seem to do anything. Do you have any advice for getting past this segment?

Gotta use the boost, but hold your grapple on that central block (little tough to grab it since you have to throw he hook at around the same time as when you're starting the boost) - if you never release the grapple and it's an appropriate length, you can spin around the block repeatedly (windmilling around like that isn't required, but it may help with the timing, and it's kinda fun).  If you release at the right time, you can fling yourself upward and to the right.

(The main idea is that you have to use the grapple to redirect your booster momentum)