I haven't quite understood this concept yet. What do I need manpower for in aspects of colonizing stars and constructing ships?

Also, how do I get manpower?

They give 20% bonus damage on ship weapons when full. 

You get manpower from a proportion of food production:

Manpower is otherwise the amount of ground troops defending your planets. Having a lot would mean your enemy has to spend a lot of turns on invasion, or besieging for several turns with special manpower-loss modules like Titanium A2S Slugs.

Thank you. is there an average, ambivalent tactic to have all them "a bit" fulfilled`? This seems to me as a bit too much micromanagement.

Exotic rations is your friend and some food production.

What is exotic rations? A tech?

Yup! It comes along with the research tech for the upgraded version of one of your tier 1 hulls. I believe your tier 1 attack.

What determines how quickly your fleet restocks in manpower? Is there a way to increase it (aside from Hero skills)? For instance, disbanding to the hangar or being in an influence circle, etc.

I'm pretty sure some heroes have abilities that "increase manpower refill rate". Other than that, I'm not positive.

Have a look at this explanatory post by dev team:

https://www.games2gether.com/endless-space-2/forums/66-game-design/threads/22537-update-1-manpower-overview

Here: https://www.games2gether.com/endless-space-2/forums/65-general/threads/23703-questions-after-update-2?page=1

I made some questions about MP dubious situations like combining with Biofuel plant, comes with devs response.