RAID card w/1x mini-SAS connector : how do I physically connect 16 disks?
Yes, this is where SAS expanders, enclosure backplanes and breakout cables come in. You can obtain a breakout cable that will provide 4 ports from the SFF-8087 mini-SAS connector. You can also try to use something like the HP SAS Expander to provide more SFF-8087 ports for your use from the single RAID controller. A detailed forum about the HP expander is available here.
I mentioned backplanes only because some have embedded expanders that allow them to accommodate more than 4 or 8 drives. You're probably not working with a system with a SAS backplane, so disregard that part and focus on expansion at the RAID card level.
Stay away from the MegaCrap cards. LSI bought them for who-know-what-reason and should have put them out of their misery long ago. Get one of LSI's own SAS HBAs (assuming you want something good and from LSI, other manufacturers are out there too).
There are "SFF-8087 to 4x SATA" cables around to plug such cards into 4 SATA drives. You can find them on eBay for $5.
It's a SAS HBA at it's heart. So you can use a SAS Expander board (or backplane) to attach more SATA drives to that one HBA. The SAS Expanders simply have 1 "upstream" port and multiple "downstream" ports, pretty simple stuff.