"Must not have provided more than half of their own support during the year". What exactly does this mean when it comes to claiming my child on my taxes?

If your child had income of $5000 and you contributed $4999 toward his support, then he has paid more than half of his own support. In this case, you could not claim him as a qualifying child. he could still potentially be a qualifying relative, but he could not have earned more than $4050 for 2016.