"Right Into Your Eyes" generally refers to making direct eye contact with someone. Eye contact is a powerful form of communication that can convey a range of emotions, such as confidence, interest, sincerity, or even attraction.
The meaning of this song talks about the way you see the person's feeling right into their eyes, but they cannot express it by some unknown reason.
https://www.youtube.com/watch?v=Urx-OoNCXkI