From Fate/Grand Order Wiki
What is Skill Reinforcement?
Skill Reinforcement increases the level of a Servant's active skill. Once reinforced, a skill will gain a level and its effect will be amplified. You can reinforce a skill via the Skill Reinforcement (サーヴァントスキル強化) option in the Synthesis menu.
What do I need to Reinforce a Skill?
To reinforce a skill, you need items based on the class of the servant and the current level of the skill you want to reinforce. Low-level skills require Shining Gems, mid-level skills require Magic Gems, and high-level skills require Secret Gems. When you have enough items to reinforce at least one skill, applicable Servants in the Servant selection screen will have a message in yellow over their icons, as seen in the right-hand image.