At its simplest, a whole lot of statistics calculations; of certain parts of images probably being certain things.
Apple has substantial machine learning research online:
https://machinelearning.apple.com/research
AFAIK, Apple hasn’t documented the technical details of Visual Look Up, though general info and related is available, as are the available application programming interfaces (APIs):
On-device APIs - Machine Learning - Apple Developer
Here is a list of what can purportedly be classified; a list of results the statistics might (correctly or otherwise) infer:
httphttps://pysource.com/2020/10/20/create-a-simple-image-classifier/
Apple has documented more about the statistical recognition of people.
Here are papers from Apple research describing details of how people are recognized:
https://machinelearning.apple.com/research/face-detection (2017)
https://machinelearning.apple.com/research/recognizing-people-photos (2021)
The statistics sometimes don’t always work as intended, either:
https://www.theguardian.com/technology/2021/mar/08/typographic-attack-pen-paper-fool-ai-thinking-apple-ipod-clip
As happened with Microsoft Tay, “Adversarial training” is a thing, too.
There are other image classifiers around from other sources that are documented:
https://pysource.com/2020/10/20/create-a-simple-image-classifier/
Yes, the term “machine learning” within computer programming is arguably another name for some forms of what mathematicians call statistics. With hardware, and side-effects.