What does the blinking red dot mean on a key fob for a Toyota Prius 2007?

It's normal for this red dot to flash a few times when you press the button normally. That's because the system is checking the key. But what you said about it only flashing when you get close to the vehicle is an abnormal situation.
I had a similar problem after I replaced the battery with one that wasn't an original factory battery. I guess the vehicle couldn't recognize my key. After I rematched the key later, this problem disappeared.

 

It means the fob and the SKS in the car are communicating with each other.

 

It also indicates when you need to replace fob battery, which is when it gets dim or doesn't light up at all...

 

It means the key fob and the automobile are communicating and hand shaking with one another and acknowledging that everything is a-okay and the correct remote is around the car and doors can be opened and unlocked car can be ready and everything is just like it's supposed to be Go away from the car for a few minutes and the remote will quiet down and all of that You can see that by watching the red light when you go in and sit in the easy chair and watch TV a few minutes you'll look over at the remote and it will no longer be hunting for the car and vice versa till you go out in the morning everything is perfectly normal.