How can I make this script only kill others, and not the one clicking the button?

You need to use the if statement on line 3 to determine whether or not 'players's Name is equivalent to the one clicking the button.

But this isn't going to work just on the client due to FilteringEnabled. You have to setup a RemoteEvent in ReplicatedStorage to tell the server to execute the code.

Use FireServer from a LocalScript and use the OnServerEvent event from the server to recieve the FireServer cal.

Script in ServerScriptStorage:

local re = game.ReplicatedStorage.ClickedButton --This is your remote

function onClick(c)
    for _, players in pairs(game.Players:Players()) do
        if players.Name ~= c.Name then
            players.Character:BreakJoints()
        end
    end
end

re.OnServerEvent:Connect(onClick)

LocalScript:

local re = game.ReplicatedStorage.ClickedButton --This is your remote

function onClick()
    re:FireServer()
end

script.Parent.MouseButton1Down:Connect(onClick)

function killOthers()
    for _, v in pairs(game.Players:GetChildren()) do
         local name = game.Players.LocalPlayer.Name
         if not v.Character.Name == name then
             v.Character.Head:Destroy()
        end
    end
end