It's one of those popular little political analogies, comparing the United States to late Republican Rome, or alternatively, to Imperial Rome.
While there certainly are some interesting parallels, especially in the relative political, military and economic dominance of the world,* it's easy to get carried away.
Firstly, it's highly debatable whether the US is in decline at all. Every generation feels its crisis are unprecedented and that things were better in the old days -- moral rot and all that. Now, I won't argue that everything is great in the USA. Some things are better and some things are worse, but we should also remember that the good old days including things like lynchings, disenfranchisement, sexism, racism, civil war etc. If there's truly a decline going on, we're not well-placed to spot it. I think the decline was very well along in Rome before people recognized it.
And Rome was different in many critical ways from the US. This doesn't make the US immune from decline, but it does suggest that it will decline in different ways and from different causes. Personally, I think analogies to Britain's experience may be more on target.
Among the differences that I think are critical is that the US is a federal republic, not a city state that acquired and empire. There's necessarily a different relationship between the parts and the whole. The US is, geo-politically, an island and naval power, secure from physical invasion. Rome was a land power, with very long and indefensible borders. Despite regional differences and ethnic politics, the US is a remarkably homogeneous society and culture, while Roman culture, while widespread, was often confined to the elites and did not displace local cultures in many places. Finally, technology and cultural development have created a far different world from 2,000 years ago, making analogies questionable. To the extent that human nature has not changed, you can learn something from Rome's experiences, just as Plato's Republic can be mined for insights today. But too much as changed to make that your primary source material. There are no steppe barbarians to keep at bay.