Unfortunately, the cladogram is not an image so I couldn't figure out how to post just the cladogram, so you will just have to go to the page.
The common names are noted on the right side of the chart and include monkeys, lesser apes, great apes and humans.
They are saying that to be consistent, everything that shares a common ancestor with Simiiformes should be called "monkeys" since old world and new world monkeys are at the base of that clade, so monkeys are paraphyletic. And everything that shares a common ancestor with Hominidae should be called "great apes" but they aren't, we called humans "humans", so apes are also paraphyletic.
It is the traditional terminology that makes the clade paraphyletic.
So I would say this: Humans are not monkeys or apes, they are "humans." They are however, in the clade Hominidae which also includes the great apes. They are also in the clade Simiiformes which includes the monkeys, lesser apes and great apes. They are also in the clade Primates, etc.,
Whoever calls me ignorant shares my own opinion. Sorrowfully and tacitly I recognize my ignorance, when I consider how much I lack of what my mind in its craving for knowledge is sighing for... I console myself with the consideration that this belongs to our common nature. - Francesco Petrarca
"Nothing is easier than to persuade people who want to be persuaded and already believe." - another Petrarca gem.
Ignorance is a most formidable opponent rivaled only by arrogance; but when the two join forces, one is all but invincible.