Link Skills are automatically triggered when characters sharing the same link skills are placed beside each other. At least two characters are needed to trigger a Link Skill. Linked characters will have the bonus effects due to the effect of the Link Skill. If there are multiple Link Skills that applies to the linked characters, the effects will stack. Linked characters will have some sort of aura emanating from their portraits.
To learn about the character's link skill, access Team Formation, tap and hold the character's portrait to bring up the Status screen. Tap the arrow in the bottom of the character's page to see his/her skills. If you expand the details of the Link Skills, you'll be able to bring up a list of all Linkable characters for the selected character.