What does "improve them twice as much" mean with respect to smithing perks?

The Smithing perks are a touch misleading.

First, consider this major point:

You don't need any smithing perks to improve any (non-magical) piece of armor. If you have the right material, you can improve it at the appropriate place. This will temper your weapons to (e.g.) (Fine) quality descriptor.

The smithing perks let you be more effective with your tempering at lower smithing skill, so instead of upgrading your gear to (Fine), you can upgrade it to (Superior).

The UESP has compiled this handy table to explain the effects of tempering.

Fine -> Superior -> Exquisite -> Flawless -> Epic -> Legendary

This is the order of upgrading you can do with smithing. Each perk allows you to access the next tier upgrade much earlier than you normally would be able to. The Legendary/Epic tiers can ONLY be accessed if you have the perk corresponding to the weapon/armor you are trying to raise.

Also, I blew all my perks on the left side, thinking I could go all the way around to pick up daedric and dragon smithing and still be able to make myself light armors on the way...turns out, you can't work backwards from the top, meaning if I want daedric smithing, I'll need to use FOUR MORE perk points on smithing, rendering the entire left side useless (as dragon scale armor is better than glass and can be made regardless of which side of the tree you raise; similarly, daedric is better than glass).