The claim you put as your title is a bit misleading, since the article itself is only talking about Red 3, one food dye used for coloring. Not all food coloring necessarily causes cancer, but Red 3 might.
The article claims that "when a substance is shown to cause cancer in animals, it is presumed to cause cancer in humans." According to the FDA website on Red 3, "the way that Red No. 3 causes cancer in animals, specifically rats, does not occur in humans so these animal results have limited relevance to humans." This is fairly vague wording, but presumably means that Red 3 may not be carcinogenic for humans. The FDA page also explains how the use of Red 3 for foods and ingested drugs was already authorized before Red 3 was "associated with animal carcinogenicity." We may be able to infer that removing authorization for use of a chemical is difficult which is why the use of Red 3 for cosmetics and topical drugs was denied, but it can still be used in food.
The source is Center for Science in the Public Interest, a non-profit organization that advocates for safer and healthier foods, so it most likely does not have alternative intentions that would drive it to provide false information.
Consumer Reports claims that Red 3 has actually been proven to have negative behavioral effects on children and that the reason the FDA hasn't taken action to remove Red 3 from the accepted list is because of complicated internal processes. This information is probably true.