I can understand why the phrase must be really hurtful and dismissive. I know that childfree people care about children just as much as parents. And while my counter point below may still feel hurtful, please know I am trying for it not to.
It's worth noting that people who are saying this have existed as both childfree people, and people with children. They have the point of comparison.
What they mean is that when you have a child, suffering of other children takes on this new dimension where you involuntarily feel a vicarious sense of that danger happening to your own children. It creates panic, vulnerability, the desire to protect your children (probably biological). It's quite different to the empathy that all good people have towards children suffering.
I work in a field (criminal law) where unfortunately the suffering of children comes up a lot in different ways. Across different sectors which work together (police, social workers, criminal prosecution and defence) people often move away from crimes involving children once they have their own. Is it because they care more? No- and in fact they're now helping those other children less, by leaving that area. It's because the way child harm impacts you as a parent is different and relates to your own children.
Of course, there are parents who probably do think they are "more" caring about children than childfree people. They're idiots using a biological protective instinct as proof of a higher morality.
I hope that this contribution did not feel patronising and understand if people still disagree.