Why do fractured parts act like an explosion? In the following we are testing cell fracturing applied to a UV-sphere (remeshed to blocks, BTW). We then apply rigidbody physics to the generated parts.  

When we start simulation as you can see below it looks like an explosion! Why? Secondly parts are continuing to bounce for long time even though their bouncing property is zero. To avoid parts going out of camera we put a passive rigidbody (big sphere). It's bouncing property is also zero. Why they act like that and how to solve this? For all parts and big sphere the collision-shape is mesh and friction value is 0.5.









Update:
Convex-hull collision, all shards are convex
  



Convex-hull collision, NOT all shards are convex
  



Mesh collision, all shards are convex
  



Mesh collision, NOT all shards are convex
  

For the above simulations:
Split Impulse is checked, Steps Per Second=120, Solver Iterations=50.

As can be seen from the above simulation results:


if all shards are convex, convex-hull collision results in the best.
if not all shards are convex, mesh collision results in the best.


Not sure why it is like this, after all!



Update:
We incorporated all the suggestions below you can find in the answer and comments, but it seems there are lots of difficulties that are not solved yet, after all! Look at the following simulations where an spherical shelf is fractured. None of convex-hull or mesh collision-shape works correctly. Check the animation in the right, it looks it is not a shard but a airplane!! This is because of use of convex-hull for shards. Check the middle one (mesh collision-shape for shards) they don't act like shard but like spring! :(
Basically it happens because when it simulates, the solver thinks meshes are intersecting with each other and it tries to separate them asap, giving the shards (sometimes) big speeds. Already mentioned Split Impulse might work and should in most cases, but there are couple more things you could try:


In Scene -> Rigid Body World increase Steps Per Second and Solver Iterations for more accurate simulation.
Try changing between Convex Hull and Mesh in Physics -> Rigid Body (for each simulated object) and adjust the Collision Margin parameter; it is not said which works better for any scene, Convex Hull simplifies the simulation so it may speed up the process, but mess the simulation even more,
Increase Margin when performing cell fracturing - this adds some space between the shards and may help a bit with simulation,
Scale whole scene up and apply scale for all objects - usually simulation performs much better for bigger objects; if you use this method, you might want to increase simulation speed in Scene -> Rigid Body World.


To copy parameters for all rigid bodies use Rigid Body Tools, select all objects you want to set parameters (like Margin for example) and copy from active Rigid body object (the one you have selected last).