"Misty-eyed" is a description of a person's eyes just before they cry. There are tears in their eyes but those tears have not yet started to roll down their cheeks.
I haven't been able to find a decent photographic example.
This is the closest I've found, especially the picture on the left. In the one on the right, the tears have started to fall and she is wiping them away, but in the one on the left, the tears have appeared in her eyes but not yet fallen.
The phrase is used because when you have tears in your eyes, your vision becomes a little blurred, as if there is a mist in front of your eyes.