How can I modify DataStores with Workspace parts?

Just because your function is connected to an event (like a ClickDetector's MouseClick event) doesn't mean that the code will run any differently. This works the same as any other code that uses IncrementAsync, although you might want to use variables outside of the connected function to avoid sending a datastore request on every single click.

An example of using variables outside of the connected function so that you don't need to send constant requests would be:

local clickDetector = workspace.something.clicky
local dsService = game:GetService("DataStoreService")
local timesClickedDS = dsService:GetDataStore("TimesClicked") -- obviously "TimesClicked" doesn't matter one bit as long as its consistent
local playerService = game:GetService("Players")

local timesClickedTable = {}

clickDetector.MouseClick:Connect(function(player)
    if timesClickedTable[player.UserId] then -- if there's a value  already saved
        timesClickedTable[player.UserId] = timesClickedTable[player.UserId] + 1 -- increment counter
        -- maybe something else
    else
        timesClickedTable[player.UserId] = 1 -- set to one since clicked once
    end
end)

playerService.PlayerRemoving:Connect(function(player) -- when someone quits
    local pUserId = player.UserId
    if timesClickedTable[pUserId] then -- check how many times they clicked this session
        timesClickedDS:IncrementAsync(pUserId, timesClickedTable[pUserId] -- increment it to their saved data
    end
end)

playerService.PlayerAdded:Connect(function(player) -- for each player
    while true do
        wait(300) -- every 300 seconds
        if timesClickedTable[player.UserId] then -- if there's any data
            timesClickedDS:IncrementAsync(player.UserId, timesClickedTable[player.UserId]) -- save it to their datastore
        end
    end
end)

In this particular example, someone would be able to autoclick to increment the counter a lot more than (probably) intended, as there's no debounce or anything that will change once they've clicked it once.

As I'm pretty sure you've figured out (judging by your question), you can find a lot of information on the wiki page for datastores.

Edit: Since you wanted an answer where the value is available between scripts, you can adapt the original code to use IntValues instead of script-specific variables. This wouldn't require much change, but I'll repost the whole thing edited as the changes are spread throughout the script. Changed lines will be marked with "--***--" above them.

local clickDetector = workspace.something.clicky
local dsService = game:GetService("DataStoreService")
local timesClickedDS = dsService:GetDataStore("TimesClicked") -- obviously "TimesClicked" doesn't matter one bit as long as its consistent
local playerService = game:GetService("Players")

local timesClickedTable = {}

clickDetector.MouseClick:Connect(function(player)
    if timesClickedTable[player.UserId] then -- if there's a value  already saved
        --***-- added .Value twice so that it updates value of object rather than replacing the reference to the object in table with a number
        timesClickedTable[player.UserId].Value = timesClickedTable[player.UserId].Value + 1 -- increment counter
        -- maybe something else
    else
        --***-- next 4 lines are all new to set up IntValue
        local intValueSetup = Instance.new("IntValue") -- make an IntValue to store data
        intValueSetup.Name = "timesClicked" -- name it so it's easier to access from other scripts
        intValueSetup.Value = 1 -- start at 1 because it was 0 and they just clicked
        --***-- using intValueSetup instead of 1 since it's an object now
        timesClickedTable[player.UserId] = intValueSetup -- put value in table for later
    end
end)

playerService.PlayerRemoving:Connect(function(player) -- when someone quits
    local pUserId = player.UserId
    if timesClickedTable[pUserId] then -- check how many times they clicked this session
        --***-- added .Value to check the IntValue's value rather than saving an object (doesn't work)
        timesClickedDS:IncrementAsync(pUserId, timesClickedTable[pUserId].Value) -- increment it to their saved data
    end
end)

-- in case of server crash or forced manual shutdown
playerService.PlayerAdded:Connect(function(player) -- for each player
    while true do
        wait(300) -- every 300 seconds
        if timesClickedTable[player.UserId] then -- if there's any data
            --***-- added .Value to check the IntValue's value rather than saving an object (doesn't work)
            timesClickedDS:IncrementAsync(player.UserId, timesClickedTable[player.UserId].Value) -- save it to their datastore
        end
    end
end)

As you can see, it is largely the same aside from adding .Value to check the IntValue's value and extra setup for the IntValue since it's an object not just a number. If there's any confusion, I'll likely edit the post later.