web site states the advantages - incl continuity of operations – stabizes situation; passporting fintech services – not to worry about regulation changes; receiving funds, payments in euros – managing currency fluctations; digital single EU market – digital signing reduces admin costs; etc
what e-Residency is doing - you would never need to visit the EU/US to establish location independent business + you wouldn't need a middleman + through digital ID you can access all the other services also. Eventually, the more players in the network the more everybody wins.
fyi, how to apply: 1.april Estonia opened foreign embassies where you can give paper application. But from May Estonia opens online application where you can pay by credit card. And once you got accepted then you need to physically go and collect the card from your chosen foreign embassy. everything here https://e-estonia.com/e-residents/about/
Why one should? yes, access to EU market, access to good services (google, paypal, soon stripe etc), access to e-banking, and generally digital signature gives hassle-free business environment (signing contracts with partners, filling e-tax declarations, administrating business on a business portal).
Why Estonia is doing it - to increase its economic reach / gain more customers / PR
I guess the second important aspect is that the other person, who you share with, is a real person. He cannot be fake as to get this ID card you need to authentificate yourself with biometric information