From the Editors of The Daily Cat
That's partly true. Cats do become anxious when stared at for any length of time. In the wild, this can be a sign of aggression or intimidation that may ultimately lead to a big catfight. Looking away is preferable to a kitty stare-down, but here's something even better to try:
Look into your cat's eyes and then blink slowly. Pet your cat and offer verbal praise after you do this a few times. If your cat also blinks at you by batting both eyes shut, feel flattered: It's a sign of affection and comfort in the feline world!