I’ll take a shot at explaining in detail, if you need more information.
Rounds:
Like you said, when units fight, the combat takes place over several ‘rounds’. Each round, the units’ relative strengths (attacking and defending) determine the odds that one of them will be ‘hit’ that round. The simplest example is a battle between two unpromoted units of the same type, on flat terrain: Each would have 50% odds of winning each round. Winner takes no damage that round.
When a unit loses a round, it takes damage. A healthy unit can absorb 100 points of damage. Depending on the relative strengths of the units, a round of damage could be, say, 15 points (or 20 or 11 or 17 or some other number… it all depends on unit strength). If a unit takes 15 points of damage per lost round, it can lose 7 rounds before dying, for example. Most battles last until one of the units is dead (or withdraws, in the case of siege units and flanking units). That could take between about 5-15 rounds, usually. You can see the record of the combat rounds by opening the Combat Log window (CTRL-TAB). Animated unit graphics can also help you visualize this – most melee, archery and gunpowder units are represented with 3 animated people, siege and mounted units with 2 siege machines/horses/elephants, ships, planes and vehicles with one machine, etc. With each round of attacks, you may see one of the 3 animated men/ 2 horses collapse and die, or the ship/vehicle representing the unit become progressively more tattered. Those animated deaths/damage represent a lost combat round or rounds. When the last animated figure collapses or the animated vehicle blows up/collapses/crashes/sinks, that shows the unit has lost the battle, obviously.
(parenthetically, once you’ve played the game a while, you can predict which unit will win the combat while the combat is ongoing, just from observing the order in which animated figures die)
First strikes:
First strikes are rounds at the beginning of a combat where the first-striking unit can take no damage. So when your Oromo fought the Axeman, it would be invulnerable for the first 4-7 rounds of the battle. Each round the Oromo wins, the Axe would take damage, but the Oromo would take no damage in rounds the Axe won. Big advantage for the unit with first strikes!
Once those 4-7 rounds are up, the battle proceeds normally. 4-7 rounds of taking no damage is a big advantage, but if the Oromo lost all (or most) of those rounds, it wouldn’t benefit (much) from that advantage. And after that, the Axe would do normal damage to the Oromo (in rounds 8, 9, 10, etc…).
My guess is that the Axe had some defensive advantages – maybe it was behind walls or on strong terrain, or it was heavily promoted – or the Oromo was already damaged, which would give the Axe good odds from round to round (a healthy Oromo would have odds much better than 64% against an ordinary Axe on flat ground). Consequently, the Axeman won most of the early ‘first-strike’ rounds where your Oromo was ‘invulnerable’. But after those rounds were done, the Axe kept winning, and your Oromo eventually succumbed to the cumulative damage from rounds after the first strike rounds were over.
TL;DR: First strikes work really well when you have stronger units facing weaker units, because your stronger unit is more likely to emerge unscathed and be able to attack again without pausing to heal. If the defending unit is stronger through promotions or other defensive bonuses (like the Axeman in your example), first strikes may not help as much as a straightforward Combat or Shock promo would.
