Influence used to work for galciv one like this:

Basically your influence meant how much sway your culture has over the population (Yours or your enemies) having high influence may make it possible for the colonies belonging to your enemy to switch sides and join your civilization.

Industry refers to production, I believe it is just a calculation of your manufacturing power as it can be displayed on the minimap. Higher industry means the planet can produce more stuff.

Approval as it is now doesnt seem to affect game play much, that will probably change. It may cause your party to loose an election if you have an advance government. And it will make your planets more vulnerable to informational warfare when you are invaded by the enemy,c ause large number of your populous to side with the invaders.