The reason debt matters is because if it gets high enough, it may cost us more to borrow money and could eventually trigger a debt spiral where we're unable to service our existing debt. Currently, our debt to GDP ratio is among the highest in the world [1][2]. At the moment, we're still able to borrow cheaply mainly to the dollar being the reserve currency of the world, and due to the petro dollar program (where oil producing companies sell their oil in dollars), but these things may not last (which is part of the reason why the US is pushing for war with China).
The takeaway from all of this isn't necessarily that we need to cut spending on social programs while maintaining a larger military than the next several nations combined, which is what the deficit hawks referenced in this article want, it's that we should rollback the tax cuts that led the deficits in the first place. We should also be increasing domestic investments that would make us more attractive to outside investors rather than becoming an over-militarized hollowed out shell of a former super-power.
If you treat your domestic economy and workforce like a leveraged buyout target, siphoning every ounce of profit out of it while saddling it with debt, the end result is likely the same as with many leveraged buyouts: collapse.