I recently got this chain master-link for my 21 speed bike but it turns out that it's really meant for a 24 speed drivetrain. Does this mean that the master-link won't be good on my set up? Or that it'll still work just fine? I really don't want my bike to be affected or the shifting to get harsher.
Like so many other things in life, size does matter.
You need to have the master link matched to the chain size (which depends on the number of speeds and manufacturer (which is basically campagnolo versus everyone else)). For example, a master link for a 11 speed chain will be too small for a 7/8/9 speed chain.
Manufacturers generally suggest you use their chain closing mechanism (Shimano's special rivets, KMC/SRAM's master links, Campagnolo's special rivet that needs peening, etc.), though many people mix and match master links across brands without issue.
In this case, you'll be fine -- 7/8 speed bikes use the same chain (you have a 3x7; the 24 would be 3x8), so you can install the link on the chain.