Slavery in the modern world can happen for two reasons. One of these reasons has to do with the supply of potential slaves and the other has to do with the demand for them.
One reason slavery can happen is that there are people who are willing to sell themselves or their children into slavery. This comes about largely because of the amount of poverty that exists in the world and the fact that people in the developing world are often desperate to come to the rich world where they hope to have better life chances. Sometimes, such people willingly and knowingly sell themselves or their children into slavery, which they expect will only last for a number of years before they are freed. Other times, they are tricked, thinking they are going somewhere for a legitimate job, and are then forced into slavery.
The other part of the equation is demand. We in the rich world want all sorts of services and we want them to be cheap. The same goes for rich people in the developing world. This leads to a demand for slaves. Slaves can do things like domestic work much more cheaply than paid workers. They can be made to do things like working in restaurants, again providing that service for much less than a legitimate worker would. Because people who have money want many things at low prices, slavery can continue.
Thus, slavery can continue to happen today because there are people who are desperate for work and because there is a huge demand for cheap goods and services.