There are two concerns. First is that China develops a nuclear deterrent on par with US and Russia. This traditionally means a second-strike capability that can guarantee that if a country with this capability were destroyed in a strategic exchange, the country who started it would share the same fate.
A second-strike capability (against ABM defenses) can be done simply by producing "ordinary" ICBM's, and launch platforms, in numbers large enough to overwhelm defenses. Given the balance of industrial capacity between US and China, and demonstrated ability to land things on the moon and mars etc, there it is a not a question of ability, but one of financial cost. With hypersonic weapons, it would be cheaper to get the same end result.
The second equally important concern, is that additional options open up by having a weapon type that can defeat ABM and air defenses singly, rather than by virtue of overwhelming numbers. This would make it possible to respond proportionally to possible US "kinetic" actions, that would be very inflammatory but short of total war, without limiting the possible responses to ones that are highly likely to produce full escalation.
"Hypersonic weapons" referenced by this question are maneuverable, fly with lower trajectories, can hit targets with precision better than that required for strategic nukes, and most importantly, can avoid ABM and air defense systems individually, rather than relying on safety in numbers. All ICBM's are technically hypersonic in terms of speed alone, that by itself isn't sufficient.
From the point of view of China, the first use-case for this technology is to be able to respond to US making "limited" attacks on mainland China, such as a conventional attack on a key target like a port. The ability to make a proportional (non-escalatory) retaliatory strike on the continental US would take this threat off the table.
The second use-case for this technology, from the point of view of China, is to take out naval targets, as well as land bases for naval aviation, at extremely long range, covering much of, or even all of the Pacific Ocean. The context for the above is that it would increase the cost and risk of the US attempting a naval blockade of China.
The third use-case, from the point of view of China, is that it would make it possible to respond to the use of tactical nukes, again without having to escalate to a world-destroying MAD event.
It is believed in some quarters, typically by those who place a high value to unchallenged US "primacy" in use of force, that each of the above would "reduce the credibility" of the US position in international discussions.
We should also bear in mind that all of these scenarios are very unlikely. It is partly because of very generous budgeting, that there is an entire industry of planners and analysts and who work full time to write papers about the subject and publicize their findings, so it can generate unnecessary excitement.