Scaffolding is an educational principle where the teacher provides the minimal amount of support to help a child learn a new concept. Enough support so that the child stays engaged and doesn’t give up, but not so much that the teacher takes away the child’s feeling of ownership (“I did it”). If the teacher steps back too far, a learning opportunity is jeopardized, because the child feels overly challenged and underprepared.
Bridging social interactions is very much the same concept. Stepping in with too great of a presence takes opportunity to gain valuable social skills away from the child. Stepping away too much potentially disrupts the likelihood that the interaction will get off the ground at all.

Here’s a silly example drawing:

T has a toy car. J sees the toy and begins to approach. It is known that T responds to unknown approaches with the following behaviors – puts his head down with no verbal response, ignores the child and walks away, becomes exceedingly protective of his things and alienates/scares new child off.
Knowing these likely responses, your bridge has to be more involved in the introductory interaction. Get up and do a play-by-play – “T. I think the boy is really interested in you and your cool toy car. I bet he’d love it if you showed him how it works”. Or, “hey J (other kid). I’m Anthony, T’s dad. It looks like your about the same age as T. Do you like toy cars also? T, what kind of car are you playing with?”.
Often this brief interception is a chance for modeling (live action) an initial interaction in front of your child and bridging the insurmountable gap between “hey” and play. What I regularly see following my interjection is that the conversation or the play takes off…”I have that car…mines the fastest…you want to race?”. Now suddenly we have an extended social interaction that otherwise wouldn’t have occurred without the bridge.