The reason of the damage has not yet been identified, but there were reports of seismic activity near Malta around teh same time the outage started.

The cables have not been completely cut, but have been damaged to a degree affecting a high number of the individual fibers within the three cables. France Telecom provides further details about the effect of this outage onto connectivity between the different regions in a press release:

Most of the B to B traffic between Europe and Asia is rerouted through the USA.
Traffic from Europe to Algeria and Tunisia is not affected, but traffic from Europe to the Near East and Asia is interrupted to a greater or lesser extent (see country list below).
Part of the internet traffic towards Réunion is affected as well as 50% towards Jordan.
A first appraisal at 7:44 am UTC gave an estimate of the following impact on the voice traffic (in percentage of out of service capacity):
-    Saudi Arabia: 55% out of service
-    Djibouti: 71% out of service
-    Egypt: 52% out of service
-    United Arab Emirates: 68% out of service
-    India: 82% out of service
-    Lebanon: 16% out of service
-    Malaysia: 42% out of service
-    Maldives: 100% out of service
-    Pakistan: 51% out of service
-    Qatar: 73% out of service
-    Syria: 36% out of service
-    Taiwan: 39% out of service
-    Yemen: 38% out of service
-    Zambia: 62% out of service

France Telecom immediately alerted one of the two maintenance boats based in the Mediterranean area, the “Raymond Croze”. This France Telecom Marine cable ship based at Seyne-sur-Mer has received its mobilization order early this afternoon and will cast off tonight at 3:00 am with 20 kilometers spare cable on board. It should be on location on Monday morning for a relief mission.
Priority will be given to the recovery of the Sea Me We4 cable, then on the Sea Me We3.
By December 25th, Sea Me We4 could be operating. By December 31st, the situation should be back to normal.

The outage has been picked up by a number of news websites. Here are some examples:

BBC News: Severed cable disrupts web access
Bloomberg News: Severed Cables in Mediterranean Disrupt Communication