One of the main reasons that they call Las Vegas 'Sin City' is because people commit sins when they are in Las Vegas.
With all of the gambling opportunities, people are sometimes irresponsible with their money. They will spend more money than they actually have and to many people, this is considered to be a sin. Some people consider gambling to be a sin and since they think gambling is a sin, then Las Vegas would be the sin capital of the world because of all of the gambling opportunities that are available in Las Vegas.
Another reason that people call Las Vegas 'Sin City' is because there are illegal activities that take place there such as prostitution, drug rings, and underage drinking and gambling. All of these activities are considered to be sins to many people.
Therefore, Las Vegas is often referred to as 'Sin City'. As they say, "What happens in Las Vegas, stays in Las Vegas".