Understand how DEFCON levels are assigned. In this case, a DEFCON 3 was issued because of the possibility that any errors in the resulting standoff might trigger open war across the Korean border (a politically and militarily tense area then, as it is today).
For instance, during Operation Paul Bunyan, which resulted in a DEFCON 3, two American officers were killed by North Korean forces in the Korean Demilitarized Zone (DMZ). Historically, DEFCON 3 has usually corresponded to situations in which military action against the US or one of its allies was a distinct possibility.In addition, all military communications may be encrypted based on classified protocols.
At this level, American military forces are put on a heightened state of alert in anticipation of mobilization - in particular, the Air Force is kept ready to launch operations with just 15 minutes of notice. Situations that call for DEFCON 3 are serious - though they may not pose an immediate threat to the existence or stability of the American state, they do call for significant vigilance. Use DEFCON 3 for tense military/political situations.