Multi-strike works exactly like the tooltips and cards describe it: the character gains X extra attacks on that turn.

Assume there is an attacker with multi-strike 2 (let's say you used 2 emberstarter and gave them unstoppable and in defense there are 2 invincible defenders. In this case, the interaction with unstoppable would work like this: If the character kills the defender with a single hit, he will proceed to attack the next one because of unstoppable. If he can't kill it in one hit (which he can't), he will attack him again, thus using one of the extra attacks. If that second attack kills the defending character (which it does), the attacker proceeds to attack the second one due to the unstoppable effect. Since he has used his multi-strike, the emberstarter will only land 1 hit on the second defender and stop attacking. The second one will finish the defender of and, due to unstoppable, will attack the fortress twice, since it still has his multi-strike intact.

That should sum up every possibility of interaction.

TL;DR: Multi-strike gives a character extra hits that are not consumed by unstoppable's effect (which gives an additional +1 attack if the criteria is met, aka dead character)