Tested this promotion with tanks last night. The tank with first strike had one of those little sidebar comments (upper left, where it tells you who won in red or green) saying that it had used its first strike while attacking. So it's not just a defensive bonus.
Not 100% sure, but it appears to me that it gives your unit the ability to hit the enemy first in that particular round. So instead of simultaneous hits I get the first shot, then the weakened enemy hits back. By this standard I'm guessing two first strikes means you get to do it for two rounds before it goes to simultaneous hits.
Some clarification in the rules or Civ-o-pedia would have been nice.