It indicates how much you used an item. Familiarity boosts both attack/defense and the additional affixes on your armor and weapons. Divine and Ethereal Gear can reach up to 999 Familiarity.
You gain familiarity by using the equipment in question. In general, skills give you less familiarity per hit then quick and strong attacks. You can however increase the gains in familiarity you get by infusing your armor and ranged weapons with +% Familiarity, using Idokuri Guardian Spirit and by using Uchiko Powder and Living Weapon. Living Weapon in particular boosts the amount of Familiarity you get.
The amount of familiarity you get also increases with the amount of amrita a killed enemy is worth. The factor is really small and estimated to be around 0.00000065% of amrita dropped by enemies. On lower difficulties, this will result in no additional familiarity gained. On Way of the Nioh, this will however severely impact the amount of familiarity you gain.
The only items that can directly increase your familiarity are Whetstones and Nikawa Glue. Your Familiarity will increase by 150 for every Whetstone/Nikawa Glue. Whetstones can be crafted from Kappa Shells, and Nikawa Glue can be crafted from Slender Vertebrae.
The Axe Method:
Equip a really weak Axe infused with Paralysis, Close Combat Ki Reduction, and Ki Damage. Use Itokuri and any other +% Familiarity Bonus equipment and enter Marobashi. Pop Uchiko Powder and use Mad Spinner on Marume Nagayoshi. He doesnt dodge and will continualy be unable to move during mad spinner, due to the paralysis. You keep spinning until you have max familiarity or until he dies.
The Water Sword Method:
Enter Hyakki Yagyo, use a weak Kusarigama/Dual Swordsas much +% familiarity bonus equipment as you can, move to the back of the spawning Yokai Bosses and Water Sword them to death.
The Living Weapon Method:
Stack familiarity bonus and +%amrita gained, use uchiko powder, extraction and pleiades and activate a Living Weapon with a high Action and Recovery stat, and kill big Yokai. This only works on WotN, because it's main focus is not on hitting the enemy repeatedly, but about benefitting from the Living Weapon enhanced Familiarity gain, and the 0.00000065% amrita dropped mechanic. This method essentialy utilizes an infinite living weapon build - and is therefor the most usable. Ranged Weapons can also be used during Living Weapon, and therefor also benefit from the enhanced familiarity gain during Living Weapon. This makes Soul Purge Talisman somewhat viable in WotN when running a LW build.