The reason you should care about your IMEI being blacklisted is that it can lock you out of your phone. Imagine being on a road trip, and your phone suddenly stops working because it's been blacklisted, not fun! You might not even be able to make emergency calls, which is a bit scary.
Another reason to check your IMEI is that it can affect your phone's resale value. If you're planning to sell your phone or trade it in for a new one, a blacklisted IMEI can reduce its value significantly. It's like trying to sell a car with a bad history, it's not going to fetch a good price.