I still haven't found time to explain this in the guide, but the only reason there are two of the boots there is that Mobafire sells the sixth-to-last item you built to make a new one. The real idea is: Phil, Boots, HoG, Sunfire, FoN, GA, sell Phil for Rylai, sell HoG for... Something. Right now, I'm thinking either Frozen Heart or Abyssal Scepter, but you'll have enough money for both, so I threw the expensive one on the list. The last 3 items will be situational, and you might want to go for upgrading the moneymakers, Warmogs, or more AP, or even Thornmail instead. Or a number of other things I can't think of right now--give me a few days, I'm tired.

Before I get to it, here's why I chose Rylai:

1. It gives this build a moderate amount of extra health, which it's a little low on.
2. It gives you AP, which I find nice and just hard to itemize for on Amumu. In conjunction with Abyssal, it gives Amumu a decent offensive. Abyssal's MPen is handier than you think because it increases his incidental damage from stuff like Sunfire, and it helps out a whole lot in teamfights; if you're considering it, getting AP from elsewhere would be nice.
3. The slow helps a fed Amumu start soloning enemies, and making sure they don't get away. If you want to spring for a Rylai's early, it might even make your ganking ability sing.
4. Teamfights! Remember, Amumu is the initiator! He sets up pins so you can knock them down. Constantly slowing everything around you falls under the category of "Setting them up". With Rylai's slow constantly affecting the enemy team, your teamfights will be one-sided massacres, 'nuff said.

The moneymakers are in part viable because they can each be turned into solid mid-late game items. But by that point in the game, I tend to have a decent enough economic status to sell them and get whatever I want.

P.S. I didn't bother with the Rylie either

Overall the best build I've used for amumu. I made 1 tweak to it, which is the kinda obvious one. upgrading the Philosophers stone into Shurelya's Reverie after getting the Frozen Heart and leaving my items at that. I can tank an entire team during the team fight stage and make a few people rage quit with all the times I've turned certain death into a kill or assist. Timing is everything with amumu and that come's with practice, time you get from staying alive with the massive amounts of resist you have X3.

2 the same boots? and why relais?