Happiness provides 1% production value increase (additive) per point over 100 to all your working kittens. This will also increase the catnip consumption of your working kittens by the same amount.
Happiness also provides 1% production value decrease (additive) per point under 100 to all your working kittens. This will not decrease the catnip consumption of your working kittens by the same amount.
Amphitheatres and Broadcast Towers can reduce your amount of unhappiness. Each Amphitheatre will reduce your unhappiness linearly by 4.8% of your current unhappiness percentage. This reduction is subject to diminishing returns, which will come into effect when your unhappiness reduction exceeds 75%, ie when you have 16+ Amphitheatres. Broadcast Towers are much more effective, reducing unhappiness by 75% each, meaning that diminishing returns will come into effect as soon as you have more than one.
Choosing the Fascism Policy eliminates the penalty to happiness caused by high population entirely. This does not affect the penalty caused by overpopulation.