So they can attract students whose parents are searching for a school.
The Summer Lunch Program (which has a much broader target audience/geographic area) in Seattle has experimented with online advertising, but had poor results. The keys to the kingdom really are reaching kids (and their parents) where they are, in schools, after school programs, and perhaps some online platforms (though the targeting is generally ineffective on Facebook/Google for this type of advertising).
I'm assuming they do both. I don't know about the US, but around here Catholic schools accept non-Catholics as well, and those may not be connected to the church.
I went to a Catholic high school, despite my family being basically non-practicing Presbyterians. And in an area with a ton of private schools.
At the time public schools were a road to nowhere, and private schools were easily $20,000 in today's dollars, maybe $30,000 depending on the school. Catholic high school cost $8000 in today's dollars, and the only "downside" was a couple of mandatory masses.
At least it had support for the learning disabled.