SNI removes the requirement from HTTPS websites to host one domain per IP. This is done by having hostname part of the initial handshake.
Many websites or at least most (hopefully all) webapps these days serve over https, at least the ones that require user input like login. So there's the need for at least one public IP per site.
With SNI these could be all served from the same IP.
Now imagine CDN services like AWS CloudFront -- to support sites with SSL certificates you must use one IP per cert in EACH region or whatever the distribution granularity is.
Now this my friends is why AWS CloudFront asks $600/mo for each custom SSL cert domain and with SNI custom SSL it costs $0/mo.
As soon as all the 99.9% browsers/clients support SNI we'll be living in a better place with more free IPs. So we can finally distribute static content from CDN through custom SSL without the $600/mo pricetag.
It's naive to think everyone would go to SNI-based hosts with IPs shared with strangers but at least within the same datacenter for the same company IPs can be more easily conserved.
Why? A router could still offer private IPv4 and encapsulate IPv4 packets in IPv6. The carrier can decapsulate IPv6 packets and perform NAT.
In fact, this is how my home cable connection works (per DS-Lite [1]). Our modem/router only gets an IPv6 address, but IPv4-only devices work fine.
[1] http://tools.ietf.org/html/rfc6333