Why I see so many clicks just after sharing a link in Facebook or Twitter?

When you post a link in Twitter or Facebook, the website publishes an instant feed about the recent tweets or updates.
This feed is accessed and crawled by various bots/spiders like Google bot to index/track the new content.
If you check the reports in your ClickMeter account, you will see that time-stamps of these clicks are very close to one another and concentrated within few minutes from the moment you posted the link.  Hence you will see a peak of clicks just after the sharing.
ClickMeter record some of these clicks as bots/spiders but cannot recognize all of them, so they may appear as human clicks.