The .bin and .cue file are the parts of the actual game disc that you want. The .bin file contains almost all of the data and the .cue file contains some extra information about the structure of the CD. All the rest is Internet Archive stuff (and an image of the game cover of course).

To open it, you can convert it to a .iso disk image instead, which any Linux distribution can open as if it were a real CD. This blog post talks about how to do that. The last paragraph about mount you can probably replace with double-clicking the .iso file in the GUI I would guess.

the bin and cue files are a cd/dvd image. IIRC you can't mount those directly, but you can convert them to iso with bin2iso (there are probably other tools too)

iso file you can mount something like mount -o loop /path/to/my-iso-image.iso /mnt/iso and then pull the files out from there.

As for directly pulling files out from bin/cue.. dunno.

I don't know if this applies to CLAW, but many games back then had their audio stored as CD Audio Tracks. If that is the case, you might want to actually emulate a CDROM drive instead of just extracting the files. There is a CDROM emulator for Linux, called CDEmu, which can read CUE/BIN CD Images.

Oh, and that game seems to have an ancient 16-bit installer, which might not work on modern systems. However, according to WineHQ Appdb one can just copy the files from the CD and it works.