How do scientists know the age of the Moon? ...of the Earth?

The age of the earth is estimated as 4.54 ± 0.05 billion years. This is based on evidence from radiometric age dating of meteorite material and is consistent with the radiometric ages of the oldest-known terrestrial and lunar samples.

Radiometric dating has been understood and used for more than 100 years and is consistently verifiable.

The age of the Earth is estimated from the age of Moon rocks returned by Russian automated probes and Apollo missions and from the age of meteorites, all of which have been determined by a variety of radiometric methods including two different uranium-lead methods, thorium-lead, samarium-hafnium etc. While some difficulties remain, examination of tungsten isotope ratios indicate that the Earth is some 30 million years older than the Moon and that the impact hypothesis is essentially correct, though the impacting object may not have been Mars size with a glancing blow as most popular accounts suggest, but smaller and a direct hit.

Return of rock specimens from other planets would go a long way to making the estimates better.

they don't, astronomy scientists think they're smart but they aren't

Radio-isotope dating. They went around the world and looked for the oldest rock, and decided that was the age of the Earth. They also dated moon rocks and meteorites.