The press release carrying the quote titling this post - "An estimated 7.7 million children in the United States (16.5 percent) have at least one mental health disorder" - concerns the findings reported by Daniel Whitney & Mark Peterson . Their research letter discussed findings (from the United States) "providing recent national and state-level estimates of the prevalence of treatable mental health disorders and mental health care use in children."
Based on data derived from the 2016 National Survey of Children’s Health (NCSH), a "nationally representative, parent-proxy survey of US children younger than 18 years" that has been mentioned more than once on this blog (see here and see here), researchers present some important data. Including information from over 46 million children (now that's what I call a decent sample size) various trends were observed, notably that almost one in seven children and young adults were reported to have a mental health condition. Such conditions covered "depression, anxiety problems, or attention-deficit/hyperactivity disorder" and by present, I mean that parents responded in the positive to the question: "Has a doctor or other health care provider EVER told you that this child has” a mental health disorder?"
Another detail was also mentioned in the Whitney & Peterson paper: "half of the estimated 7.7 million US children with a treatable mental health disorder did not receive needed treatment from a mental health professional." This was based on responses to the question: "DURING THE PAST 12 MONTHS, has this child received any treatment or counseling from a mental health professional? Mental health professionals include psychiatrists, psychologists, psychiatric nurses, and clinical social workers." Other media on the Whitney / Peterson paper have picked up on this trend (see here) and the possible whys-and-wherefores.
The primary weakness of the NCSH - "parent-proxy survey" - is more than compensated for by the huge participant numbers included for study. The figures arrived at also follow a trend seemingly present across many nations (see here and see here and see here) suggesting that significant numbers of young people are experiencing mental health disorder. We can quibble about the reasons for the increase but there is no mistaking the fact that something is going on. And it's seemingly affecting millions of children and young people around the world...
And as if to prove the point further  the startling findings from Gräf et al: "School performance was available for 1462 children (51% boys, mean age 7.3 years). Of these, 41% had signs of at least one MHP [mental health problem]."
 Whitney DG. & Peterson MD. US National and State-Level Prevalence of Mental Health Disorders and Disparities of Mental Health Care Use in Children. JAMA Pediatrics. 2019. Feb 11.
 Gräf C. et al. Mental health problems and school performance in first graders: results of the prospective cohort study ikidS. European Child & Adolescent Psychiatry. 2019. Feb 26.