After you have fought, you will receive a battle report. This will be found together with the rest of the game notifications, in the Military Messages screen. You can find this as Messages in the Military game menu. When you have chosen a battle report to read, just click on it.
Battle reports are usually very simple. They contain a list of all the different unit types that were in the battle, and, for every player: how many units there were before the battle began, and how many were lost during the battle. In addition, for you only, there will be stats for how many units your units killed.