tighten up (on sth) means: to become stricter or more carefully

Haha funny video. To tighten the reins literally means to pull on the reins when you are riding a horse to make the horse stop or slow down. Here "tighten up your reins" means to start to control something or someone more carefully. Example: She has tightened the reins on her younger sons in an effort to curb their wild behavior before it's too late.

What an adorable video until a little more than 2 minutes into it!