Need help with Fleet Capacity

Wow are you serious? hehe.
First, you must have a least one research structure build. A military or economy research structure it doesn't matter which. Then, when you are in your game you press the button at the very top middle of your screen called Research (f2 hotkey) to bring up the upgrades. On the left click on Fleet Logistics. Here there are two screens upgrades sequences shown. The top upgrades increase your maximum capital ship size limit and the bottom upgrades increases your fleet capacity. Click to upgrade the first of the bottom line of buttons to allow for greater fleet capacity. When you use that you upgrade the next level and so on.

In the research window, there are several options on the left hand sidebar: Military Civilian, Fleet Logistics, and Artifacts. Fleet logistics has buttons that will allow you to research higher fleet capacity and slots for additional capital ships.

Be careful, though. Once you start increasing your fleet capacity, you start incurring increasing maintenance costs for your fleet. At the beginning, you don't incur maintenance at all. At the first level of increased fleet capacity, you lost 9% of all income (credits, crystal, and metal) for maintenance. So don't upgrade fleet capacity too quickly. Make sure you're actually going to use those slots you're about to pay for.