Wasn't the last scene of that episode Tywin entering the Throne room where Cersei was about to give poison to Tommen? Pretty sure I remember that...
Point is, we can only assume how both battles were won, nothing was shown. It is clearly not in the interest of the writers to make a part of this show about the battles, their tactics and strategies. The most we ever get to know to logically predict who should win it is the number of soldiers and in some cases how strong they are, which is possibly the highest for the Unsullied.
In case of Mereen, she sent a decent amount of the most clinical army in the continent with feck loads of weapons that enabled all citizens/slaves to be a part of the attack. One can also assume there was a big factor of surprise as the masters were not expecting their slaves to suddenly turn up with swords, deadly ones. And so far what we have seen in both continents, if you kill the person/group controlling the city army, the army simply joins your forces and doesn't fight to protect the city. When Melisandre's shadow son killed Renley, Stannis simply acquired those men and ships, so you can assume something similar happened with whatever army mereen had, if it did. I don't think it did, because in the last episode during the small council meeting they only talk about the Unsullied and the Second Sons. So possibly the city was run by masters who enslaved people and had no army, perhaps since there was no immediate threat or perhaps in a case of war they would have had the money to buy other armies, which wasn't the possibility in case of an inside invasion.
Basically the point is the war/taking over/conquering aspect is hardly detailed at all and it goes for both continents. Everytime there is a war it basically focuses on the main characters and leaders and the defeat/assassination of the leader means war is over, doesn't matter who had the stronger or bigger army. So it's just that in Dany's case almost all her storyline revolves around conquering cities and this lack of detail hurts her a lot more than the ones in the westeros.