My dh did this recently. Never thought he'd find out his dad as no name, one night stand, birth mother now dead although they met.
First linked on Ancestry by a distant cousin who joined the dots and gave him names. His dad is sadly dead and never knew about him. He has an uncle and is tentatively finding out stuff. A shock for him I'm sure, he had no idea.
I very much take dh lead on everything. Try and keep an open mind as to outcome, good or bad. His meeting his mother was good and so sad it was limited by her death, it could easily have gone another way.
Make no mistake it will turn peoples lives upside down.... those that have mentioned surely its water under he bridge in terms of infidelity, rape or abuse, after so many years are spectacularly naive.
DHs birth mother's whole family's idea of her would have changed. She had to tell her new husband about a child from years before that she never did before, she wasnt the person he thought he knew. She never told her parents (religious) who are still alive and only some siblings who she felt would understand. Imagine your spouse or family member telling you something like that that had shaped their lives many years before. I'd be shocked if my dh or siblings or children had done similar.
Other pp examples of their experiences have shown me how lightly we have come off through this. I couldn't imagine the pain of being rejected for him if that had happened nor the stress of a birth parent who suddenly demanded all his attention and expected to be the primary parent in his life or similar. He has only 1 half sibling who seems ok now he knows but is so much younger and in another country with language barrier, we have no idea of his true feelings which will surely be tied up with the death of his mother when he was still a child.
Its complicated and no right or wrong way to approach. I'd just be there for your dh to listen, mentally prepare for bad news or reactions and welcome positive steps with caution, peoples feelings can change either way after the initial emotional reaction. After very positive meet ups initially my dh and his mother needed a break and didnt contact each other for a while. She later admitted she had struggled with coming to turns with it all. They didnt get far enough to really go through all their feelings sadly, her death was a shock, we didnt know she was ill and due to language barrier dont really know the circumstances and what she or other family members went through 
I wish you all the luck for a positive outcome for you all x