The questions are answered in the article. The equivalence is not supposed to be obvious even to other number theorists. Most of the article is devoted to explaining "why" it works since the actual proof is two paragraphs.
Don't know what you're talking about. The article is about unproved theorems. They are described as being simple or obvious, but there's nothing obvious about
Prove that for n > 1,
divsum(n) < H_n + exp(H_n)log(H_n)
See the entire field of number theory for something that didn't have applications when it was conceived. Gauss, Euler and the other early number theorists were essentially just amusing themselves with stuff like the phi function. Then cryptography became a serious open problem, and boom.
Interestingly enough this is most often the case with pure mathemetics: their proofs are pursued purely in the interest of understanding & attaining them - the implications & applications of which are sometimes unrealized for decades (a great example is Evariste Galois: his number theory and symmetry/group work was decades-later applied by physicists to define many of the laws physics appear to obey - check out the wiki).
Or a certain obscure piece of algebra discovered by one G. Boole.
(Edit: During his time, Boole's results were practically unknown except to other logicians, and certainly nobody expected to find any practical use for them. It wasn't until the 1930s when Claude Shannon realized that the algebra that now carries Boole's name could be used to analyze digital electric cirquits.)
A novel proof involves making novel connections, rigorously showing how different parts of mathematics can be made to work together in ways that haven't been thought of. When you look at it that way it's not hard to imagine the value in solving puzzles like this, even if it's impossible to know the specifics ahead of time (which are by nature unknown).
