Does the internal CD-ROM drive's tray open and did you try the system disk in it? I'm wondering why an external optical drive has been connected to the computer, unless the internal drive stopped reading disks. Pre-OS 8.1 Macs require that any burned data CDs be formatted as HFS (not HFS+) volumes (explained below), so that could be the problem with the installer disk that only spins and isn't readable. If it's a burned disk, it must be on older/slower speed CD-ROM media, been burned at a slow speed, been formatted as HFS, and been created to be bootable. Old versions of Toast provided these options. I suspect that more recent versions have dropped the HFS formatting option. In my experience, those older 4x SCSI CD-ROM drives often had problems reading burned media, even the latter 8x and 12x SCSI drives. I also found that CD media with an aqua or turquoise surface color were more likely to be readable than CDs with a mirror-like appearance.
OS 7.6 was Apple's last operating software made available on a floppy disk set. By the time OS 8 was released, the number of floppy disks needed for the installation was increasing and the number of users installing from floppy disks was decreasing. The external optical drive is (as Jan stated) a SCSI CD-ROM drive. Apple's first DVD-ROM drives were ATAPI devices. Above or below the Centronics-50 cable that's plugged into the back of the SCSI drive, you should also find a (preferably "active") terminator plugged into the other Centronics-50 port. If there's nothing plugged into the 2nd port, is there a slide-switch on the back to enable termination? If the end of the SCSI chain isn't properly terminated, the computer will have startup problems. Also, the DB-25 SCSI port on the back of the 6320 is connected internally to the same SCSI bus as the internal CD-ROM drive is connected. This means that the external drive can't have the same SCSI address as the internal CD-ROM drive uses: Address 3. There should be a button on the back of the external drive to change the SCSI address. The 6320's internal 1.2 GB hard drive is connected to a dedicated IDE bus.
Prior to OS 8.1, hard drives were restricted to being formatted as HFS volumes (Mac OS Standard). As hard drive capacities kept increasing, the minimum allocation block size kept increasing and created wasted space, wherein small files were swimming around in oversized allocation blocks. The Mac OS 8.1 Update introduced the HFS+ format (Mac OS Extended), which addressed this issue, the same way that FAT-32 did for PCs. While a Performa 6320 can run a more modern OS than 7.5.1 or 7.5.3, the hardware is taxed when trying to run OS 9.x. OS 8 or 8.1 would be the best compromise for the feel of a modern OS, and with a 1.2 GB internal hard drive, the issue of HFS versus HFS+ isn't a big deal. The 6320 originally shipped with 16 MBs of memory, probably a pair of 8 MB SIMMs. If your 6320 hasn't had memory upgrades and you wanted to run OS 8 or 8.1, I'd remove one of the 8 MB SIMMs and replace it with a 32 MB, for a total of 40 MBs. If you stick with OS 7.5.x or 7.6/7.6.1, 16 MBs will be adequate.
If the motherboard's original 4.5-volt alkaline battery is dead and has leaked solution on the motherboard, corrosion could be an additional cause of problems, so removing the motherboard for a visual inspection would be first on my checklist to troubleshoot. Loss of the stored PRAM settings from a dead battery is often a source of startup issues. Be sure to discharge any static charge that you may have acquired, before getting your hands near the components on the board. The Performa 6320 has a slide-in motherboard that's removed from the rear. If this computer has been in storage for a while, you should examine the upper and lower surfaces of the gold-plated pins along the edge of the motherboard. Discoloration or oxidation can compromise any one of a number of functions, because that edge connector engages a wide slot that connects the motherboard to the floppy, optical, and hard drives, video port, as well as to the power supply. If discolored, I gently scrub the pins with a soft, white eraser to restore a shine. Following that, any rubber shavings must be completely blown off before installing the board.
