Why won't my local script change the background color smoothly transition into white?

In order to do this without it going out of bounds, you will need to use a few checks. You do have the basics down though.

local color = script.Parent.Background.BackgroundColor3 --variables are fun, use them
local r = script.Parent.Background.BackgroundColor3.r
local g = script.Parent.Background.BackgroundColor3.g
local b = script.Parent.Background.BackgroundColor3.b

for i = 1, 255 do
    wait(0.1) --now that it is simplified you only need 1 wait()
    g = i --if i is going to increase from 1 to 255, might as well assign the g and b values to i so you don't have to do extra math.
    b = i
    print(r.. g.. b)
    color = Color3.fromRGB(r,g,b) --while it is true that you could just write 255 instead or "r", I'd still use the variable r in the case that you decide to change this in the future.
end

Now if the r/g/b values are all different numbers, it gets a bit more complicated, but not by much:

local color = script.Parent.Background.BackgroundColor3 --variables are fun, use them
local r = script.Parent.Background.BackgroundColor3.r
local g = script.Parent.Background.BackgroundColor3.g
local b = script.Parent.Background.BackgroundColor3.b
local red = false
local green = false
local blue = false
for i = 1, 255 do
    wait(0.1)
    if r ~=255 and not red then
        r = r + 1
    else
        red = true
    end 
    if g ~=255 and not green then
        g = g + 1
    else
        green = true
    end
    if b ~= 255 and not blue then
        b = b + 1
    else
        blue = true
    end
    print(r.. g.. b)
    color = Color3.fromRGB(r,g,b) 
    if red and green and blue then
        break -- the red green blue check allows it to stop looping early if it reached a full white before the loop was finished.
    end
end

This should work for you. it worked for me. I don't really know how to explain why yours wasn't working, though it should be fixed. It kept printing out a lot of weird decimals. I don't know much about R G B, so I don't know why it was printing out a bunch of crazy decimals. Maybe you weren't adding 1 to the value the proper way. I think that is why yours was broken, so I added them using a different method. Here you go.

    local g = script.Parent.BackRound.BackgroundColor3.g
    local b = script.Parent.BackRound.BackgroundColor3.b
for i = 1, 255 do
    wait(0.1)
    g = g + 1
    b = b + 1
    print("255 ".. g.. b)
    script.Parent.BackRound.BackgroundColor3 = Color3.fromRGB(255, g, b)
end

You can use i but there is a library/plugin you can use to do this.

You can use the RoStrap plugin to tween many properties that can’t be tweened using Roblox’s TweenService and it is also smoother. I’d recommend getting the plugin and reading the documentation to easily tween color.