Okay, so let's start with what "illegals" means. It's a word some people use to talk about people who came to a country without following the official rules.
Now, the "illegals program" is a term for different programs designed to help people who are living in a country without papers. These programs can vary a lot depending on the country, but they usually aim to give some kind of legal status to these people so they can work, study, or be less afraid of being sent back to their home country.
Some people think that everyone should follow the rules and there shouldn't be any programs for people who came illegally, while others believe that everyone should have the same opportunities, no matter how they arrived in the country.