Many characters in League of Legends use a resource called "Mana" to power their spells. Evelyn, whom you're playing in the picture above, is one such champion.
The abilities show up in blue because you do not have enough mana to activate them. The simplest solution is to return to the fountain, where your character will rapidly recover both health and mana.
In terms of longer term strategies to avoid running out, there are plenty of available items in the store that increase mana, mana regeneration, or both, and the Ancient Golem (aka "Blue Buff") will rapidly refill a percentage of your resources as well.