Why is this Fireball script not working?

Skip to the bottom if you just want the working code, but if you want to understand why your script didn't work, read the analysis below. I explain things that aren't necessarily errors as well, so some things will work, I just prefer a differing method. I changed some variables to make them shorter, sorry.

Use local whenever you define a variable (not when you call it again since it's already defined). It generally helps the script lag less, though the effect is minimal unless you call the variable a lot. It's just a good habit to get into.

The character will presumably die (especially if you're throwing a bunch of deadly fireballs around) so you'll need an update for characters of players. The first character won't cut it!

You have a boolean that keeps the player from spamming fireballs, except that you make it false as soon as the function runs, so it's essentially useless. The other two portions of the boolean are ok, and definition was ok.

You are making this ball fly at the speed of sound; I doubt anyone can even see it. I personally find Body-userdata things and the Touched event to be unreliable. Because of this, I tend to CFrame everything that moves. I know a lot of people like Lerp. I've never been a fan; I just write out something similar.

You lack a kill script. This is the part where I say "You're giving the client too much power" because I'm a FilteringEnabled fanatic. Personally, I would add a RemoteEvent for this thing. You'll see what I mean in the final code.

Now, you could make this code even more efficient than what I did by having the client project the object since it will be less of a burden to the server, with the server giving position values for the ball, but that isn't really what you asked for, so I'm going to give you the object being spawned in Workspace by the server.

Add this portion to the Server Script.

local cDownTab = {}
local range = 100 -- This is in studs.
local speed = 90 -- This is in studs per second.
local fps = 20 -- This is how many times your fireball position is
-- updated a second. Higher values may cause your speed lag, especially 
-- when more balls are in-game. You can solve this by making 
-- graphics local but keeping game logic server controlled.
local intialTrans = .4

function checkFireBall(obj,plrN)
	if obj then
		for _,v in pairs(game.Players:GetChildren()) do
			if v.Name~=plrN and v.Character then
				local torso = v.Character:FindFirstChild("Torso")
				local humanoid = v.Character:FindFirstChild("Humanoid")
				if torso and humanoid and humanoid.Health>0 and (torso.Position-obj.Position).magnitude<obj.Size.X/2 then
					humanoid.Health = 0
				end
			end
		end
	end
end

game.Workspace.Caller.OnServerEvent:connect(function(plr,arg)
	if arg[1] == "Fireball" and arg[2] then
		local plrN = plr.Name
		if not cDownTab[plrN] then
			if plr.Character then
				local humanoid = plr.Character:FindFirstChild("Humanoid")
				local torso = plr.Character:FindFirstChild("Torso")
				if humanoid and humanoid.Health>0 and torso then
					local Ball = Instance.new("Part")
					Ball.Parent = game.Workspace
					Ball.Shape = "Ball"
					Ball.Size = Vector3.new(5,5,5)
					Ball.BrickColor = BrickColor.new("Really red")
					Ball.Transparency = intialTrans
					Ball.TopSurface = 0
					Ball.BottomSurface = 0
					Ball.Anchored = true
					Ball.CanCollide = false
					Ball.CFrame = torso.CFrame*CFrame.new(0,0.5,-5)
					print("Fireball started.")
					local vectorCalc = (arg[2].p-Ball.Position).unit
					print("Arg", arg[2].p, "Pos", Ball.Position, "Calc", vectorCalc)
					coroutine.resume(coroutine.create(function()
						checkFireBall(Ball,plrN) -- Check its initial position.
						for i = 0,range*fps/speed do
							Ball.CFrame = CFrame.new(Ball.Position)*CFrame.new(vectorCalc*speed/fps)
							checkFireBall(Ball,plrN)
							wait(1/fps)
						end
						for i = .1,.9,.1 do
							Ball.Transparency = intialTrans+((1-intialTrans)*i)
							wait()
						end
						Ball:Destroy()
					end))
					cDownTab[plrN] = true
					wait(3)
					print("Fireball successful!")
					cDownTab[plrN] = nil
				end
			end	
		end
	end
end)

Add this portion to the Client Script, like in the StarterPlayerScripts.

local plr = game.Players.LocalPlayer
local mouse = plr:GetMouse()

mouse.KeyDown:connect(function(key)
	if key:lower() == "f" then
		game.Workspace.Caller:FireServer({"Fireball",mouse.Hit})
	end
end)

You will need a RemoteEvent that looks like this in Workspace in order for this code to work. Hope this helped, and if it did, don't forget to upvote!