TL;DR: Anything you say outside the family circle weakens your position and is strenght given away to your enemy. Knowledge is everything, and Sonny gave away important information. You don't show any interest, anytime. What Sonny did was a big mistake, and Vito instantly understood that. He shows anger. And he's probably disappointed by his son's attitude.
He even says to Sollozzo:
I have a sentimental weakness for my children and I spoil them, as you can see. They talk when they should listen1
In that video, look at the faces of Clemenza and Hagen. They, too, understood the wrongness of Sonny's intervention. It's an interference. It's undermining the authority of the Don. No one in charge is to be cut off like that. An organization where the boss can be interrupted when he talks is a weak one, seriously impaired, a mess, with no hierarchy. It's a weak place waiting to be conquered by a competitor.
When Sonny did that, it's like saying: "get rid of the boss, he's old and doesn't understand what should be done, he's missing a great opportunity. I'll be in charge and we'll make the deal". That's exactly what Sollozzo did afterwards, when he tried and have Vito killed. Sonny opened a door, they stepped in. Never tell anybody outside the family what you're thinking.
Vito knew that. He made a 'friendly' statement when saying 'no' to Sollozzo. He just gave one reason: losing more than winning. It's just business, nothing personal (so that you don't offend the person). Often (wrongly, I think) attributed to Sir Winston Churchill, there's a saying (roughly translated, I couldn't find it in English) that states you should:
Make your decisions public, not your reasons. Decisions can be good, reasons will probably be bad. (William Murray ?)2.
Sonny's mistake was understood by everybody but him. Sonny's hot temper leads to terrible things from his father's point of view: he talks too much, has a mistress ("What's the matter with you? I think your brain is going soft with all that comedy you are playing with that young girl."); and remember that scene when he talks to Michael and says that a man should always take care of his family and spend time with them. He always reacts to events instead of thinking ahead3. That's why it was so easy for Tattaglia/Sollozzo to trap and kill him. After his death, Vito even says once (if IIRC) "Sonny was a bad Don, RIP...".
All of these mobs have choosen a violent path, where the sentence is almost always the same: violent death, from an enemy. You protect your family, and information is power. You need people in the police, in the streets, in offices, politicians and so on. You need to know, without disclosing what you know. They usually are not afraid, but they don't want to expose themselves. Be very quiet and don't show off. See how, at the weeding (in the movie) Barzini has the photographer ripped off of his camera and takes. No evidence. No information. Ever.
To me, Vito doesn't show fear, but he already understood that Sonny's blunder will lead to moves against the family. He may not know which ones, but there will be. The simple 'no' he opposed Sollozzo's request is enough to put them in danger, because he knows that, behind Sollozzo and Tattaglia, there's Barzini, the smartest of them 3, and that they want more money and more power. That means, at the Corleone's expense.
Sonny lacks a few (amongst many) qualities a Don needs: he's not calm, doesn't listen, and doesn't forsee. He wants to fight, all the time, starting as when he was a kid. He's what standards mobs are: a man who doesn't care much about the others, and live his life fighting for power and money. Strenght of the arm over the strenght of the head. The gun before the brain. He gave the enemy important information. Vito uses the brain (and the threat) much more than the gun, unless necessary.
The good general wins the battle before he even starts it.
Rule is that the winning one is the one with the best information.4
To secure ourselves against defeat lies in our own hands, but the opportunity of defeating the enemy is provided by the enemy himself.5
1. excerpt from the movie.
2. William Murray
3. Keep your friends close and your enemies closer. You'll be safer if you know more about your enemies than you know about your friends. Look after your enemies better than you look after your friends. Another great advice by Sun Tzu.
4. Sun Tzu - the Art of War
5. the Art of War
From a comment by anongoodnurse:
If you go by how the media portrays Mafioso, they all seem to fool around or have mistresses (Tony Soprano, etc.), it seems to be acceptable behavior. What the Don didn't like was how careless Sonny was; anyone with eyes could see his indiscretions, which is disrespectful to their wives. They should be more careful. But Sonny isn't a careful guy.
6. Keep your friends close and your enemies closer.
Extra: Some great quotes.