I recently got my Blacksmithing skill to 100. Now in the skills menu I can press space to make it Legendary, which will put it back to a skill level of 15 and reset my perks.
I am wondering: What benefit is there from making a skill Legendary apart from being able to level up the skill more thus having infinite levels?
There are a few perks to making a skill legendary:
Since your perk points reset within that skill tree, you can reallocate those points within the tree or place them into another tree
Resetting the skill back to 15 allows you to continue levelling using a tree that you actually use in the style you play the game with instead of forcing you to max a tree that is not within your playstyle. For example, I prefer to play a stealth/ranged damage character, so don’t particularly care for tanking so I could use the legendary skill feature to remove the need to max the block and heavy armor trees.
As you stated, the legendary skill feature effectively removes the level 81 level cap, so that you can attain ridiculous levels. A side effect of the level cap removal is that you continue to gain a perk point each level. You could theoretically get to a high enough level to have enough perk points to get every perk from every tree.
Source: http://elderscrolls.wikia.com/wiki/Skills_(Skyrim)