How do I make a script give points to ONLY the person who touched it?

Really Easy.

first you need to make a leaderbored and make sure name the leader status points

For example I using points

on the first line you want to a touch function

function onTouched(part)

Secound, you need to find the player.

local h = part.Parent:findFirstChild("Humanoid") --Humanoid is the player
    if (h~=nil) then
        local thisplr = game.Players:findFirstChild(h.Parent.Name)
            if (thisplr~=nil) then

Remember that leader bored you made? Time to find it in the player!

        local stats = thisplr:findFirstChild("leaderstats")
            if (stats~=nil) then

let add a value!

if (score~=nil) then
                    score.Value = score.Value + amnt

Time to finsh up the script with connecting the mouse and let add a amount part :D

amnt = 1 --how much you get for it
function onTouched(part)
    local h = part.Parent:findFirstChild("Humanoid")
    if (h~=nil) then
        local thisplr = game.Players:findFirstChild(h.Parent.Name)
        if (thisplr~=nil) then
            local stats = thisplr:findFirstChild("leaderstats")
            if (stats~=nil) then
                local score = stats:findFirstChild("Points")
                if (score~=nil) then
                    score.Value = score.Value + amnt
                end
            end
        end
        script.Parent:remove()
    end
end

script.Parent.Touched:connect(onTouched)

That all hope this help :)

Are you asking for normal points or Player Points? If it's player points then here is what you need.

-- declare service
local PointsService = Game:GetService("PointsService")

-- Bind function to player added event
game.Players.PlayerAdded:connect(function(player)
    -- Get total number of points the game has available
    local pointsToAward = PointsService:GetAwardablePoints()
    -- Get total number of points this game has already awarded to the player
    local universeBalance = PointsService:GetGamePointBalance(player.userId)
    -- Check if the game has points to award and if the player hasn't gotten any points yet. If both are true, then give the player a point.
    if ( pointsToAward > 0 and universeBalance == 0) then
        PointsService:AwardPoints(player.userId, 1)
    end
end)

-- Bind function to when points are successfully awarded
PointsService.PointsAwarded:connect(function(userId, userBalanceinUni, userBalance)
    -- Show message indicating that a player has gotten points
    local message = Instance.new('Message', game.Workspace)
    message.Text = "Point awarded to " .. userId .. ". This player now has " .. userBalance .. " points total!"
    wait(5)
    message:Destroy()
end)