I don't understand clearly why the L2-norm regularization term forces the model to be simpler (and why the L1-norm for example doesn't)

That is incorrect: L1 regularization also yields ‘simpler’ models than its unregularized counterpart.

For a better understanding, and a comparison between L1 and L2 regularization, see also:
http://oknoname.herokuapp.com/forum/topic/229/l1-regularization-for-features-vs-l2/#c2