The guide
No labels, so no loss to minimise against a known answer. Instead you state what structure you expect and let the data fill it in, which means the assumption you build in is doing most of the work.
k-means assumes clusters are round, similar in size, and that you already know how many there are. DBSCAN assumes clusters are dense regions separated by sparse ones, so it finds shapes k-means cannot and labels outliers as noise instead of forcing them somewhere. Neither is right in general.
PCA is the odd one out: not clustering but a change of basis, keeping the directions the data actually varies along. It is the same eigenvector machinery from linear algebra, applied to a covariance matrix.