What is meant by setting the "wheel" of Dharma in motion?
You can read about the Dharma Wheel or Dharmachakra in the article "The Dharma Wheel (Dharmachakra) Symbol in Buddhism" by Barbara O'Brien.

In this article, it is stated that the Dharma Wheel is a chariot wheel and it is most commonly depicted as having eight spokes, representing the Noble Eightfold Path.

What happens when a chariot wheel is turned?

The chariot moves of course.

In this case, I suggest that it symbolizes that the chariot of humanity starts moving towards the end of suffering, towards Nibbana.