So I got a bit more clarity from my dad.
He says that the last screenshot (from above) shows everything is working as it should be. When he goes to browse the internet is when he gets a small message in the lower righthand corner. He clicks on that and gets the attached screenshot.
He has already entered his license key, but the message and this attached screenshot still come up. Interestingly enough, when he first got ESET installed, everything worked fine for about 6 months. Then the software told him his license had expired, which was odd. So he took the computer back to the repair guy, and the guy put in a different license number. Then all of this started happening. The repair guy emailed my dad the new license number (he hadn't given it to him this last time), but it seems like the program keeps asking him to put it in over and over.
I'm not sure what to make of any of this.