I found that a bit surprising, too, but I'm guessing there's a bit more nuance/context to that question than the short clip that I saw.
Regardless, there are rational reasons to provide health care to all who need it in a given community.
But PLEASE do a quick google search "do undocumented immigrants pay taxes?" before you continue.
[Spoiler: to a large extent, they do pay taxes, and since they can't claim any benefits, they are net benefit to the US treasury.]
At least for communicable diseases, the pathogen doesn't really care who it infects, so having some sort of minimal health care for everyone in the community regardless of immigration status makes some sort of sense. If you discourage a certain subgroup from getting help, they'll sit there in your community spreading it among your citizens. It might cost a lot less to make sure small problems get nipped in the bud rather than having it spread everywhere.
In urgent/emergency situations, the cost overhead of determining each patient's immigration status may be greater than just providing care. You don't want your citizens to have to jump through those hoops.
No matter how you look at it, alien workers are providing some benefit to someone by doing whatever work that they are doing. If their work is not of some value to the employer, they wouldn't be working those jobs. Having many of those workers out of the economy due to injury/sickness will cause "friction" in that economic activity (higher costs, delayed service).
You can argue that they shouldn't be there to begin with, but if those setting policy and doing enforcement REALLY believed the illegal alien workers are a problem, I would expect them to be going after those employers really hard, as that would be the easiest and most effective way to combat the problem. The fact that most policymakers don't seem to make any noise about that, and as far as I've seen, may of those rural Republican lawmakers themselves employ such alien labor, makes me think that they don't really think that it's a problem, either. As a voter, if you hear those same politicians going on and on about how horrible those illegal immigrants are, without a peep about the illegal EMPLOYERS which is the bigger part of the problem, your first reaction should be to question their motives.