Intimacy is a system that offers a wide variety of interaction with many NPCs. This will affect how the NPC interacts with you and in some instances will allow them to sell you additional items.
When speaking to an NPC you will have an option to begin a story exchange. This will bring up a new interface that shows the NPCs constellation. You will then have the option to select information about other NPCs you have met and share stories about them. If you succeed in your conversation you will be awarded with additional intimacy. After a successful story exchange you will be given an option to have a second conversation for even greater rewards, but should you fail you will receive no rewards.
Like each player NPCs have a birth constellation, if your constellation matches the NPC you will have a bonus on your interactions with the NPC. On the NPCs constellation you will see empty circles, each of these need to be filled with a topic of your choice, it is recommended that you go from least interesting to most to ensure success.
If an NPCs shows items in their shop as grayed out this is because you lack sufficient intimacy to purchase. Some NPCs without a shops will offer rare items if you gain enough, others will provide new quests, or give you new knowledge.