Blheli suite is an application used to program the esc parameters.
Most common parameter programmed is for throttle max and min. There are many other parameters that can be used to tune the esc to the users preferences.
The connection method is usually via USB using a USB linker and suitable driver. This is usually a silicon labs driver but is dependent on the mcu used.
There have been reports of issues with Windows 10, mainly around unsigned drivers, however I run Windows 10 and use the BLheli suite to program my esc's and have not experienced any issues.
You usually require a USB linker to connect the esc and program it. These are available from any good hobby shop. I use the DYS unit which I purchased from Hobbyking, although you should be able to get this or an equivalent unit from any good model shop. Here is the link to the one supplied by HobbyKing -
Otherwise you could use an arduino to do it. Oscar laing has a very good guide here
The blheli suite wiki can be found here
I have read that on some (flight) controllers (I have a few multi-rotor drones), they allow a function called pass-through where the esc are connected to the controller and the blheli suite connects to the controller. The controller passes through the data received to all esc's simultaneously. I have never had any luck with this method, but I do think it is related to the flight controllers I am using.
Hope this has helped.