Let's say you're getting the $2k tax credit per kid, three kids. You've done all the math, and have "inflated" your W4 numbers thus reducing your withholding per paycheck. At tax time you have a pretty low tax refund, for this example we'll say you got a $500 refund last year. Without the kids you'd owe thousands, but you have kids, and you don't want a repeat of 2015 when you got a $6,500 refund even though your W4 was spot-on.
Basically, in the past, with three kids you'd get $6,000 taken off at tax time (this is oversimplified, I know, just illustrating a point).
Now you're getting $3,000 per kid (assuming none are under six). That's $9,000 instead of $6,000. So you're getting even MORE taken off your taxes, right? Well, technically yes, but you're getting half of that before you even file your taxes. So of the $9,000, you get $4,500 before the end of the tax year. That means when you file your taxes, only $4,500 comes off the taxes...because you already got the first half of the tax credit.
If you "inflated" your W4 to account for the (very predictable) child tax credit, the numbers assume you'll get $6,000 taken off of your taxes...not $4,500. So in a scenario where with a properly filled out W4 you'd end up with a $6,500 refund, but you inflated the numbers so you'd end up with a $500 refund...well...now the numbers are off, and you'd end up owing $1,000 instead (difference of $1,500).
It really is a "Mustachian People Problem", hence being posted in this thread. You either reduce the W4 numbers thereby increasing the withholding (which will be balanced out by the advance payments starting in July), or you just elect to NOT take the advance payment (and maybe even increasing the W4 number a bit to reduce withholding a tiny bit, to account for the fact that you'll be getting a larger deduction/credit come tax time). Or just let it slide and see what happens; I think as long as you haven't had to pay in for X years and didn't change anything, they'll let it slide at least once without a penalty (definitely check on this, I'm not giving financial advice, etc.).