The Intel 82599 is a 2 port 10GE controller. QSFP is a 40GE connection mechanism. QSFP is a 40G connection mechanism but can operate in a mode where 4 10G lanes are broken out. Absent any other specs I assume that they're just using two out of the four. Honestly it seems like an expensive and inefficient way to produce what probably should just be a couple of SFP+ ports.
As far as what can be connected? You can find QSFP adapters that break out to 4 twinax or AOC cables (fixed length .5 -> 30 meters with SFP+ connectors). You can also get a few different fiber options which will be MPO/MTP 12-strand connectors. Depending on what the vendor supports this will likely include multimode (10GBase-SR) which can support up to 300M on OM3 and possibly some single mode options. I know of 40G optics that break to 10G LR (10KM). 10GBase-ER is rated for 40KM but I've never heard of a QSFP breakout module that would support these.
So - you need to get specs from the vendor for what PHY they'll specifically support. If we assume it's SR (not unreasonable) then you can procure standard breakouts to OM3 LC connectors. You'll have a live TX and RX for each of the two ports. To connect to another server (or a server and a workstation) cross the respective TX and RX links. It's just a back-to-back 10G Ethernet connection, after all.