An artist can obtain a Vegas residency by booking a series of shows at a particular venue in Las Vegas, most commonly a resort or casino.
The short answer is that they make the artists and resorts a lot of money!
A New Day — by Celine Dion Duration: 4+ years Venue: The Colosseum at Caesars Palace Gross earnings: $385.1 million
Duration: 8+ years Venue: The Colosseum at Caesars Palace Gross earnings: $245.5 million
Duration: 5+ years Venue: The Colosseum at Caesars Palace Gross earnings: $166.4 million
Duration: 4 years Venue: The AXIS at Planet Hollywood ross earnings: $137.7 million