the spontaneous connection

Do you ever see someone who you want to talk to but get caught in your head about the approach?

Follow the Inner-action for an interaction. 

"I want to say Hi
And It's difficult to say exactly why.
I could see us being friends
at least for a minute
sharing ideas
and laughing a lot. "

What motives could I have for this urge of connection?

A few of many facets of attractiveness. 

Laughter!
Authenticity
Creativity
Cooperation

Physical appearance

Mental frequency

Degree of flow or stress

Work

Heart openness
Communication style

Relationship Cultivation

How often do I feel impelled to reach out to a 'stranger' but hold myself back for some reason?

There's a difference between being invasive and inviting people to be. Invasion seems to happen when we are attached to expectations, illusions, or one way of experiencing life. Invitation to be true happens in being open to experiencing what is happening.

The only thing I expect is being open to the most powerful synchronicity that come from the law of the most high.