Why Dragger is not moving the model?

So you almost had it except you did make a few errors, and used a some methods that are deprecated. Instead of using Mouse.KeyDown we will be using UserInputService because Mouse.KeyDown is deprecated. So for example

local UserInput = game:GetService('UserInputService')
UserInput.InputBegan:connect(function(Input)
	if Input.UserInputType == Enum.UserInputType.KeyBoard then --Check if the input was from the keyboard
		if Input.KeyCode == Enum.KeyCode.R then --Checks if the KeyCode of the Key pressed matches the Enum KeyCode for R
			print("R was pressed")
		end
	end
end)

Also you didn't account that when you unequip the tool the script will still run. So we will use a few connections to fix this problem. So for example

local Con = nil
script.Parent.Equipped:connect(function()
Con = game.Players.LocalPlayer.Mouse.Button1Down:connect(function()
end)
script.Parent.UnEquipped:connect(function()
Con:disconnect()
end)

That is a example of using connections so that when the tool is unequipped the function is no longer connected, in short the code won't run. You also didn't account parts that aren't models and are parented to workspace. So to fix this I added a few lines

if Mouse.Target.Parent == game.Workspace then
	MouseDown = true
        Dragger:MouseDown(Mouse.Target,Mouse.Target.Position - Mouse.Hit.p,{Mouse.Target})

So all together now

local Plr = game.Players.LocalPlayer
local Dragger = Instance.new('Dragger')
local MouseDown = nil
local MB1DCon = nil
local MB1UCon = nil
local MMCon = nil
local UserInput = game:GetService('UserInputService')
local UISCon = nil

script.Parent.Equipped:connect(function()
	local Mouse = Plr:GetMouse()
	if not Mouse then return end
	
	MB1DCon = Mouse.Button1Down:connect(function()
        Instance.new("SelectionBox",game.Players.LocalPlayer.PlayerGui).Adornee = Mouse.Target
        if Mouse.Target.Locked == true then MouseDown = false print("Locked") return end
		if Mouse.Target.Parent == game.Workspace then
			MouseDown = true
            Dragger:MouseDown(Mouse.Target,Mouse.Target.Position - Mouse.Hit.p,{Mouse.Target})
		elseif Mouse.Target.Parent:IsA("Model")then
			MouseDown = true
            local list = Mouse.Target.Parent:GetChildren()
			for i=1,#list do
				if not list[i]:IsA('BasePart') then
					table.remove(list,i)
				end
			end
            Dragger:MouseDown(Mouse.Target,Mouse.Target.Position - Mouse.Hit.p,list)
		else
			MouseDown = true
            Dragger:MouseDown(Mouse.Target,Mouse.Target.Position - Mouse.Hit.p,{Mouse.Target})
		end
	end)
	
	MB1UCon = Mouse.Button1Up:connect(function()
		if MouseDown then
            MouseDown = false
            for _, selec in pairs(Plr.PlayerGui:GetChildren()) do
                if selec:IsA("SelectionBox") then
                    selec:Destroy()
                end
            end
            Dragger:MouseUp()
        end
	end)
	
	MMCon = Mouse.Move:connect(function()
		if MouseDown then
            Dragger:MouseMove(Mouse.UnitRay)
        end
	end)
	
	UISCon = UserInput.InputBegan:connect(function(Input)
		if Input.UserInputType == Enum.UserInputType.Keyboard then
			if Input.KeyCode == Enum.KeyCode.R then
				if MouseDown then
					Dragger:AxisRotate("Y")
				end
			elseif Input.KeyCode == Enum.KeyCode.Q then
				if MouseDown then
					Dragger:AxisRotate("X")
				end	
			end
		end
	end)
end)

script.Parent.Unequipped:connect(function()
	UISCon:disconnect()
	MB1DCon:disconnect()
	MB1UCon:disconnect()
	MMCon:disconnect()
end)

Hope you don't just use this script but learn from it.