So, to the real reason you clicked this guide and actually got to the guide part:

Why Sunfire? On Akshan? What?

Sunfire Aegis has an on-hit effect that most people seem to ignore these days. After it fully stacks up from being close to enemy champions, baron, or dragon, every auto attack will cause an explosion centered around you that deals damage to all enemies around you based on the item's damage as well as apply a damage over time effect that deals the same damage. This on-hit effect has no cooldown, and can be applied as fast as you can apply on-hits. This means that because Akshan attacks twice every auto attack, he naturally gets a higher benefit than most champions.

Now for the item synergies: Guinsoo's Rageblade allows you to not only attack faster, but also double up the amount of hits you do every other attack.
Hurricane is a very special and required item for this build, and the reason is simple; if you are auto-attacking at least three units of any type while you have Sunfire fully stacked, you'll activate sunfire three times, six times if you do the full Akshan auto-attack, TEN TIMES IF GUINSOO'S RAGEBLADE IS ALSO STACKED.

Now you get it.