Line 7 is the problem.

Value.Changed:Connect(function(plr)

And line 4 is the culprit

local Value = Stage.Value

The problem here is the fact that you wrongly assume that the Value variable refers to the actual property of the Stage.

This example would not work as expected:

-- Currently the IntValue holds a Value of 25. 
local int = workspace.IntValue.Value

int = 92 -- Here it's just reassigning 'int'.
print(workspace.IntValue.Value) 
--> 50, it does not change even after reassigning 'int'.

Though this would:

-- Starting value of IntValue.Value is 25
local int = workspace.IntValue
int.Value = 50
print(int.Value) --> 50
int.Value = 92
print(int.Value) --> 92

Reference vs Value

It is important that you know the difference between a reference and a value.

Reference

In Lua, for example, functions are passed by reference.

local func1, func2

func1 = function()
    print(1)
end

func2 = function()
    print(1)
end

local func1reference = func1
-- func1 and func1reference are referencing the same function now, but func2 is different

print(func1 == func2) --> false, not referencing the same function
print(func1 == func1reference) --> true, func 1 reference references func1
print(function() end == function() end) -->false

The func1 and func1reference variables both hold the same exact function. func2 is a different function from func1 even if it contains the same code. Remember, references.

Value

Though data like booleans and strings are passed simply via value.

local _string = "hello"
local _string_value = _string
_string = "gzhsiikw"
print(_string_value)

local bool1 = false
local bool1_val = bool1
bool1 = true
print(bool1) --> true
print(bool1_val) --> false 

Even though _string_value was set to _string, both variables have different memory and modifying one value does not change the other. The same happened for the boolean values, bool1_value never changed even after updating bool1.

Final product

game.Players.PlayerAdded:Connect(function(plr)
    local PlayerStats = plr:WaitForChild("leaderstats")
    local Stage = PlayerStats:WaitForChild("Stage")
    local Gems = PlayerStats:WaitForChild("Gems")

    Stage.Changed:Connect(function(new) -- this is not the player. This is the new value.
        if new % 5 == 0 then
            Gems.Value = Gems.Value + 10
        end
    end)
end)