Why can't I fire a client?

You have WAY too many if statements necessary for your FireClient(). Here's how you can do it:

local repstorage = game.ReplicatedStorage
local event = repstorage.StartTimer
script.Parent.Touched:Connect(function(hit)
    local plr = game.Players:GetPlayerFromCharacter(hit.Parent)
    if plr then
        event:FireClient(plr)
    end
end)

GetPlayerFromCharacter() immediately gets the player, and so you can use that instead of just doing it plr = hit.Parent.

Answer:

Step-By-Step explanation

(This is just to add a bit more of a description as to why it didn't actually fire.)

Problem:

• The reason it didn't work was because, player variable wasn't defined in your script.
• Another reason was, you're trying to :FireClient to an instance that is null, THERE'S no plr parameter on your script, which again leads to another error.

Solution:

Make sure you've defined the player, also try to avoid using many if statement's in your code that are not needed, as it makes it harder to debug.

Reccomendation:

Use GetPlayerFromCharacter() as DovyDas stated, because it gets the player who has made a contact with a 3D dimensional object, and is generally easier than using if-statements as the function returns a player hence the name "GetPlayerFromCharacter"

Fixed Script:

(You can use anyone of these scripts stated, as they both act and do the same thing.)

1st script:

local repstorage = game.ReplicatedStorage
local event = repstorage:WaitForChild("StartTimer")
local player = game:GetService("Players").PlayerAdded:Connect(function(plr)
local Part = game.Workspace.Part
Part.Touched:Connect(function(hit)
    if hit.Parent:FindFirstChild("Humanoid") then
    event:FireClient(plr)
    print("fired")
    end
    end)
    end)

2nd script:

local player = game:GetService("Players").PlayerAdded:Connect(function(p)
local repstorage = game.ReplicatedStorage
local event = repstorage:WaitForChild("RemoteEvent") -- change to your remoteevent name
local part = game.Workspace.Part
part.Touched:Connect(function(hit)
    local plr = nil
        if hit.Parent:IsA("Model") then
            plr = hit.Parent
        end
        if hit.Parent.Parent:IsA("Model") then
            plr = hit.Parent.Parent
        end

        if plr then
    else
            return
    end
        if game.Players:FindFirstChild(plr.Name) then
            print (7)
        plr = game.Players:FindFirstChild(plr.Name)
    end
    event:FireClient(p)
    print("fired")
    end)
end)

P.S: This was kind of rushed, so sorry for bad grammar!

Thanks, JesseSong!

If you need any further help, don't hesitate to comment, or leave me a DM!