It is simply calculated by adding together all the system related bonuses and maluses. It is expressed in percent from 0 to 100%, where 0% means mutiny and rebellion and 100% means absolute happiness. System Approval shows the happiness of the system's population. Notably, industry is not directly affected by Approval (a marked departure from Endless Legend), which makes the consequences of extreme approval shortages somewhat less crippling. Then, the empire itself has an overall (averaged from individual systems) approval rating, which affects a player's dust and science production. The bonuses/maluses are applied in two separate domains individual systems have their own approval rating, which affects their food and influence production.
Primarily, approval acts as a FIDSI multiplier low approval causes all FIDSI (except industry) to plummet, and can cause starvation, scientific stagnation, economic crisis, and political apathy. By having this system in place, on most galaxy sizes, empires will undergo an initial period of rapid expansion, but will reach a soft impediment, as further expansion causes their approval to plummet, and causes them to struggle with some new maluses that situation introduces. The primary purpose of approval is to curb a player's expansion - without this, the rate at which empires could spread across the galaxy would be limited only by their ability to churn out colony-and-supply ships, and all of the empty space in the galaxy would be filled quite rapidly.
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