How do image hosting websites stop illegal images from being uploaded?

They don't. Recommended reading: DMCA safe harbor provision https://en.wikipedia.org/wiki/Digital_Millennium_Copyright_A... and related. In other words, things get taken down by nastygrams from lawyers, not by algorithm.

Note that images may be "illegal" for approximately ten bazillion different reasons, and most of those reasons can't be determined algorithmically - from copyright infringement to pornography (famously defined as "I know it when I see it").

They usually have a "report photo" link.

I assume a human may review the reports and wields the ignore/delete/banhammer as appropriate.

If I were to build one, I'd ensure I kept a hash/perceptual hash (see phash and friends) of removed images so they'd get put in the queue/automatically rejected as appropriate.

Companies can have teams of content moderators or reviewers, whether employees or outsourced.

Good article is here: http://www.nytimes.com/2010/07/19/technology/19screen.html?_...