The most suspicious part of this code is this:

Part.Parent.Name ~= "script.Parent.Name"

Notice the quotes, and how the right side is highlighted in blue? That is text. Unless you literally named a part "script.Parent.Name", this will always be true.

If you want to compare the Name of Part's Parent and this script's Parent, then drop the quotes.

Cleaning Up

This script can be made significantly cleaner.

Tab your code correctly

Use local variables

It's very unhealthy to use global variables in a function. Every new variable should be local.

Redundant Checks

Checks like ~= nil or == true are redundant and make it harder to read. It's easier to read if enabled then -- it also hints that enabled is just supposed to be a boolean that's solely used for that purpose.

leaderstats

The object is supposed to be called leaderstats, not leaderstat. As far as I know, "leaderstat" won't appear in the players list, while "leaderstats" will. If you don't want it to appear in the players list, I would suggest picking a different name so that you don't confuse yourself/anyone else. If you do want it to appear in the players list, use leaderstats.

BS and SN check

You check that BS and SN exist on line 21... But you've already used them on line 13 and 14.

That means the check on 21 will always succeed, since if they didn't exist, you'd have gotten an 8 lines earlier.

If you are certain they exist, the check is unnecessary -- and then so is :FindFirstChild. I'm guessing you aren't certain (you usually can't be). Move the check earlier.

StealerName / BeingStolen

StealerName is only used to get a player out of Players. That means you should probably not store its name, but the player themself:

local Stealer = game.Players[script.Parent.Parent.Name]

Is this supposed to be script.Parent.Name? Or is the check earlier supposed to use script.Parent.Parent.Name?

Equivalently, instead of searching for a Humanoid, we can use :GetPlayerFromCharacter:

    local BeingStolen = game.Players:GetPlayerFromCharacter(Part.Parent)
    if BeingStolen and BeingStolen.Name ~= script.Parent.Name then
        -- is script.Parent.Name supposed to be script.Parent.Parent.Name?

Victim is probably a simpler name than BeingStolen.

BS and SN

Use .Character instead of searching in the workspace for a given name!

Debounce

You set enabled = false but you never re enable it again.

Summary

Here is what I have from cleaning this up. I'm sure a lot more can be done still

local enabled = true
function onDamage(Part)
    local Victim = game.Players:GetPlayerFromCharacter(Part.Parent)
    if Victim and Victim.Name ~= script.Parent.Name then
        if enabled == true then
            enabled = false
            print("Process started")    
            local Stealer = game.Players[script.Parent.Parent.Name] -- Is this supposed to be script.Parent.Name?
            local BS = Victim.Character
            local SN = Stealer.Character
            print("Variables successfully defined")
            if BS and SN then
                if Victim.leaderstat.Level.Value > 59 and Stealer.leaderstat.Level.Value > 59 and Victim.leaderstat.SpecAbility.Value == 1 and Stealer.leaderstat.SpecAbility.Value == 1 then 
                    SN.Torso.Anchored = true
                    BS.Torso.Anchored = true
                    local x = Instance.new("BoolValue")
                    x.Name = "Process"
                    x.Parent = SN
                    print("Process value inserted into the player model of SN. Waiting 25 seconds...")
                    wait(25)
                    print("25 seconds waiting, processing outcome.")
                    if BS.Humanoid.Health > 0 and SN.Humanoid.Health > 0 and SN:FindFirstChild("Process") then
                        print("Outcome successful, working on changing values.")
                        Victim.leaderstat.Level.Value = 1
                        Victim.leaderstat.EXP.Value = 60
                        Victim.leaderstat.SpecAbility.Value = -999 --meant to be nothing
                        Part.Parent.Humanoid.Health = 0
                        Stealer.leaderstat.SpecAbility.Value = -2 
                        print("Values chaged, respawning attacker and victim.")
                        SN.Humanoid.Health = 0
                        BS.Humanoid.Health = 0
                        wait(2)
                        script.Parent:Destroy()
                    end
                end
            end
        end
    end
    print("Process ended")
end

script.Parent.Touched:connect(onDamage)