Mining starbases increase the relevant stat of your empire (economy, research, morale, military, influence) by the percent listed. Note the bonuses are additive, not multiplicative - two resources mined at 39% will give 78% bonus, not 93% (Stat * (1.39 *)) not (stat * 1.39 *1.39).
Research bonus gives additional research output, so you get more output from the same number of buildings. Economy bonus increases the amount of money you get from the same population and tax rate. Morale increases the approval of your empire. Military increases the attack and defense of your ships.
Influence increases the ammount of influence you get from planets and starbases. It will increase tourism income, but not by a large factor. Your planets will be more resistant to flipping, and will more easily flip others.