Fire Rate is a powerful stat when used on the right weapon, as it can double the DPS of certain weapons when stacked enough. Despite how powerful of a stat it can be, it also comes with its own drawbacks. Making proper use of this powerful stat can help deal more DPS and clear enemy waves or bosses faster. That being said, on some weapons, maximizing fire rate can become extremely difficult if done manually, so binding the shoot button to both directions of the scroll wheel is recommended, although it will prevent you from using charged attacks. Note that these are my observations with the stat, so if there is something off with it, feel free to let me know.

Fire Rate reduces the time inbetween uncharged shoot inputs or shots in an automatic weapon by multiplying the interval between shots (abbreviated as i in this guide) by the difference between 1 and the decimal value of a percent. The formula seems like it is roughly this:

i * (1 - fire rate)

Note that this stat does not affect time spent charging a charged shot.

Fire Rate on non automatic weapons seems to be capped at about 15 shots a second, resulting in stacking the stat on both max tier boosts and a max tier weapon on certain fast firing weapons not nearly as effective as running a different stat in either the gun or one of the boosts.

Weapons cannot shoot again during the time the weapon is shooting a burst of projectiles, so if a weapon's fire rate is faster than its firing animation, the extra fire rate will not affect it.

Note: For weapons listed as Partially Benefits from Fire Rate, assuming the player is using tier 9 or tier 10 equipment, this means that better results are likely had if the player has either one of the boosts or the weapon itself have a different DPS, survivability, or mobility improving stat instead of fire rate, while the weapon and the other boost still runs fire rate. Results may vary on lower tier equipment.
Fire Rate interacts with different weapons in different ways. Sometimes a weapon can maximize all the fire rate bonuses, sometimes it will work on a weapon up to a certain point, sometimes it will only be impacted by fire rate, and sometimes it is completely unaffected by any fire rate modifiers. Here is a list detailing how weapons are affected by fire rate.
Acid Shot
Collider
Combustor
Cryoshot
Dual Laser
Human Minigun
Lightning Gun
Megashot
Mothership Issue
Repeater
Sawed Off
Spreadshot
BB Gun/Bee Gun
Alien Blaster
Energy Knife
Irradiator
Human Machine Gun
Photon Gun
Six Chambers/Revolver

Fire Rate has very good synergy with Crit on Hit%, as Fire Rate makes you do many hits, letting you get more value out of crit per hit. I recommend combining Fire Rate with other damage improving options, such as Double/Triple Bullet, Damage/Robot Damage, and Reload Speed/Magazine Size (when applicable) for boss damage, and/or more survivability outside of bosses, such as objective heal, life steal, dodge bullet regen, or defense.

Some weapons may interact differently between patches. Energy Knife formerly could use fire rate, but was changed so that it no longer does in patch 1.1.2, although existing weapons with fire rate retain its stat. Dual Laser seemed to have been shadownerfed that same patch so that it shoots ~25% slower, now requiring weapon specialist and all three fire rate slots for the fastest firerate.

If you have questions or suggestions, feel free to leave a comment. I may make mistakes, so if I did make a mistake, let me know.