Exactly. It had to do with slaves born in this country and/or people who were brought here legally (via the law structure of the time, while allowed slavery).

Would have applied to people who were brought here illegally, too. When slavers kidnapped people and brought them into the U.S. after it was no longer legal to import slaves (from 1808 onward the importation of slaves was illegal, but many tens of thousands were imported anyway), all of their children and descendants were U.S. citizens. Didn't matter that they were brought here illegally.

What does it say about illegal immigration?

Nothing. People who have crossed the border illegally are still fully and completely subject to the jurisdiction of the United States and the state they are in. So their kids are citizens upon birth. Just like every other kid. Just like every other kid whose parents might have committed a crime in the past, or who might even be committing an ongoing crime - as long as their parents are subject to the jurisdiction of the U.S., the government cannot impose a disability on a newborn baby based on something their parents might have done before the baby was born. Everyone born in the U.S. is born with the protections against government that come with citizenship, so that we can never have a situation where a newborn baby is devoid of those protections again.