HC-05 and microcontrollers
These types of cheap modules have a couple of different operational modes, depending upon how you want to use them. They can be pre-loaded with firmware and used (somewhat) as stand alone devices, or they can be used a serial - bluetooth bridge. The first option is why someone would "program" this module using some other controller (like an Arduino board).

Here is a decent Instructable on controlling devices over Bluetooth using an AVR microcontroller (the most common types of Arduino boards are based on AVR).

The reason most guides are for connecting with an Arduino is because those are the primary users of these modules. Most engineers would design their own circuit using the bare chips, but these modules are useful for prototyping and proof-of-concept type projects.

In any case, you will need a microcontroller of some kind on both ends - 1 to encode the SNES controller button information to send over the BLE channel, and 1 to decode that data to feed back into the SNES console. With that in mind, it makes the most sense to use these modules as bridges as opposed to stand alone devices. To do this, you need to know the data protocol for the SNES controller to the console. I found this guide, but I have no idea if it is accurate.

You don't really need to know much about how Bluetooth works to use these modules, but you do need to understand the communication protocol for the chips themselves. Here is the HC-03/05 Command Set