When you select a ship, the infocard at the center of the bottom UI bar has a picture of the ship and several icons on the left and right to the picture.

If I remember correctly, the bottom-left icon gives information on abilities when you hover it. You'll find the Psi Power amount for the ship there.

As for how Psi Power benefits each ability, the only information I'v seen so far is the purple text on each ability tooltip.

Thank you! I didn't realize that each ship had an in-depth breakdown of weapons/defense/abilities next to their picture. Looking at that helps clear a lot of things up. It also illuminates the Psi Power calculations. It appears that when upgrading an ability, the purple number is the amount improved per Psi Power. And when hovering over the chosen ability, it's already displaying the calculated increase. It seems like all the information is there, just not in the most intuitive location.

im not sure that psi actually does anything for the coronata mind control ability. it always says +0 for the chance regardless of how much power. 

basically ability power from other games like league

psi power empowers ship abilities. For example some skill has 300 dmg and 6000 range. With 50 Psi power the dmg will be 300 +100, range will be 6000 +2000. All numbers are jast an example but i hope You got the mechanic. You get psi power with researches to increase psi power, and items for planets or capships