They can use your information to open financial accounts, apply for credit, or commit other illegal activities.
If they don't have enough information about you to create a full identity, they use what they already have to access more details in scams. They make phone calls or send emails pretending to be from respected companies to get you to reveal personal information, such as passwords and credit card numbers. These scams are called phishing and vishing.
The more information a fraudster can get about someone, the more attractive that person is to them. They can also manipulate people to give them confidential or personal information. This is called social engineering.