The article exaggerates for clickbait purposes.
The actual plan had serious scientific intent, and was not to actually 'blow up' the moon, rather to detonate nuclear weapons on the moon's surface so that they could get a better look at the composition of the regolith and the moon's internal structure. The explosion itself would have thrown up a lot of debris that could have been imaged with spectroscopy. A large crater would be left behind that could be explored by rovers and probes, and seismic monitoring probes could have used the shockwaves to create a picture of what the moon looked like internally.
It would actually have allowed us to learn a lot of things that we still don't know. The actual effect on the Moon as we see it form Earth would have been very minor. A new small crater, and that's pretty much it.
The program was cancelled for two reasons. Firstly it would have involved putting nuclear weapons in space which may have been seen as a provocation by the Soviets. And secondly it would have committed the US to a very expensive and comprehensive set of unmanned lunar exploration missions, at the same time that the Apollo program was under way. It really came down to asking whether they wanted to send men to the moon, or do some big exo-geology experiments,
The sexier option inevitably won.
As to WI they had actually done it? We would know a lot more about the Moon, including whether if there are signification amounts of water trapped in the crust. This might have lead to more focused exploration of the moon later one. The cost involved may also have lead to the cancellation or even serious curtailment of the Apollo program. Though if significant resources had been found on the moon, a manned effort may have been mounted later on anyway.
A minor diplomatic incident with the Soviets may have occurred, but nothing major. The Soviet space program was pretty science focused, and they would have appreciated the intent.