Why is the __add metamethod not printing anything?

__add will not fire for childMeta[player].Stats.Kills += 3

it fires if you do tableWithMT + something

example

local tbl = setmetatable({},{
	__add = function(t, v) 
		print(t, v)
		return true
	end
})

print(tbl + "anything") -- prints true, the return defines what he + operator will do on that table

what you probably want in the example you gave is __newindex metamethod it fires when you try to set the value of an index that doesnt exists but in your example the value already exists so what we do is we create the actual table behind the screen and give them an actually empty table with an MT that does the interaction with that table so that it seems like a normal table

example

local actualTable = {
	Stats = {
            Kills = 0,
            Deaths = 0
        },
    Properties = {
        Name = "playername"
    }
}

local tbl = setmetatable({},{
	__newindex = function(t, k, v) -- fires only when you set a value to an index that doestn exist
		print("changed")
		actualTable[k] = v
	end,
	__index = function(t, k) -- fires only if you try to get the value to an index that doesnt exist
		print("retrieve value")
		return actualTable[k]
	end
})

print(tbl.newKey) -- prints retieve value and nil
tbl["newKey"] = "a new Value" -- prints changed
print(tbl["newKey"]) -- prints retieve value and "a new Value" 

tbl.Stats.Kills = 1 -- doesnt fire newindex because stats doesnt have a metatable so you have to create one for this too