For every unique liked item that is placed two squares or less adjacent to the monster, happiness is increased by 25%. (Two or more of the same item/monster only increases happiness once.) Thus four items are required to make a monster 100% happy. Note that on some islands, some monsters have five likes available, but it is not possible to make a monster more than 100% happy.
Happiness affects the rate of coin production. For each increase of 25% happiness, the coin production rate increases by 25% of the base value. Thus, if a monster is at 100% happiness, it will produce coins at twice the rate of a monster at 0% happiness. If the monster is at least 25% happy, the "rate" value on the info screen will display in green numbers. Happiness does not affect a monster's maximum income, i.e. the number of coins the monster will hold.
As a matter of strategy, it is worth considering how often you collect coins from your monsters. For example, a level-15 Potbelly reaches its maximum income in 6 minutes if it is 0% happy, and in 3 minutes if it is 100% happy. If you never collect coins from it more frequently than every 6 minutes, you get no benefit from increasing its happiness. Most other monsters take much longer to reach their maximum income, so improving their happiness increases the number of coins you can collect.
If an item is placed near a monster that makes it more happy, an upwards harp-strum noise will be heard, and pink hearts will fly out of the monster's head. Conversely, if an item is removed and the monster becomes less happy, a downwards harp-strum noise will be heard, and blue broken hearts will fly out of the monster's head.
The Unity Tree nullifies the requirement to place an item within two squares of a monster to make it happy. Items and monsters can then be placed anywhere on the island and still increase happiness. This allows for greater freedom in designing islands.
The Storage Structure will affect the Happiness of a monster if a liked object is taken into it. The same applies if the Hotel Structure is used to store a liked monster, unless the Unity Tree is bought.