Evolution Center is a 4x4 building that allows you to increase the level your mutants can achieve. Once your mutants reach the maximum level of your Evolution Center, they cannot keep leveling up until you upgrade this building.
The first few upgrades are fairly cheap, but they start getting more expensive as you upgrade it further, up to a limit of 900 per level, or an increasing amount of (the cost tends to rise by 3,000,000 for each level).
This in turn allows your mutants to upgrade their attacks and abilities (e.g. stronger Boost, Retaliate, Drain Life, Curse, Wound and Shield).