I have become a vegan, but not completely.  I have a lot to learn
It dates back to its  roots in the civilizations of ancient India and ancient Greece.
Why did the people stop eating meat?
It  was closely connected with the idea of nonviolence toward animals (called "ahimsa" in India), and was promoted by religious groups and philosophers.
Did people continue to drink milk and eat cheese, which is non violent?
 in medieval Europe restricted or banned the consumption of meat for ascetic reasons, but none of them abstained from the consumption of fish; 
Why was fish not included?  They are also animals.
Because veganism restricted the consumption of meat ,fish served as a better alternative
In history, when did people begin to believe eating meat is not healthy?
Since the during the 19th and 20th centuries