Found a technical explanation :
Actually, the submerged portion of the launch is on a column of compressed air, which gives enough boost to cut through about 150 feet of water before breaching the surface, where the rocket motors ignite.
The air is compressed by a small, contained explosion, something like the explosion in the cylinder of a gasoline powered automobile, but in this cases uses a column of super heated steam.
The inside of the missile tube is dry. After the outer hatch lifts away, a seal at the top of the tube is blasted away by a small charge. In a split second, before water can flood the tube, the shot of compressed air boosts the missile away from the sub. The momentum is sufficient to push the missile to the surface. Inside the missile, when sensors detect deceleration, the rocket motors ignite.
That gets us out of the water - but what happens next?
If you watch enough missile launches, you'll notice that many times the missile appears to be doing a bit of a dance before it finally flies true. That's owed to the uncertainty of launch conditions; sea currents, wind, variances in the column of compressed air, etc. In the equipment section of the missile, inertial controls detect the hull's deviation from course, and send correcting pulses to the TVC (thrust vector control) which steers the missile by changing the angle of thrust exhaust from the gimbaled motors.
Once on steady flight, an onboard camera starts looking for a certain star to use as a point of reference for celestial navigation -same principle as mariners used centuries ago. Once locked onto the star, corrections are made as required to keep the missile on course. Perhaps curiously, while GPS systems are carried on some vehicles, they are not part of the guidance package, for the simple reason that the GPS system, itself, might be compromised or out of service in the event of hostilities. Their purpose, where used, is to verify accuracy and refine accuracy of the other guidance instruments.
