I've heard/read countless sources explain that the Japanese used in anime is quite different from real world Japanese. However, I haven't seen an explanation of what ways it's different (at least, not beyond pointing to specific phrases/words that people in the real world rarely use, or simply saying that it sounds "unnatural"). Is there an analogous English way of speaking that is used in places, but normally sounds unnatural to English speakers? Is there a more general explanation as to how Japanese in anime is unnatural sounding? Or is it really just common phrases/words that are rarely used in the real world?
It very much depends on what anime. For example, an anime about daily life in the modern world would have generally "normal" Japanese. Conversational snippets sound totally normal, for the most part. You start to have "unnatural" Japanese when the writers, trying to make characters unique, give those people special "quirks" such as always referring to themselves in the third person. On occasion, in Japan, children will use this form of speech, using their own names when talking about themselves: "Yui likes apples very much!" (spoken by the girl Yui) another "quirk" is "gobi" or sentence ending particles. The standard word to indicate the end of a sentence is "desu" or "da" . But quirky anime characters are often given the habit of using special "gobi" unique to them. Squid Girl Ikamusume comes to mind.... she ends her sentences with "degeso". The word "geso" means "dried tentacles", or just "tentacles" if you're referring to the ones still attached to a living animal.
I learned Japanese in college, but I learn new phrases and vocab from anime all the time, and find it very helpful for that. BUT, many people can tell that I watch anime, in part because some phrases are rare, but also because most of the characters I watch are female. As a result, sometimes I unintentionally use speech patterns that in Japan are used only by women. It's hard for me to give examples of this, because the only ones I know well enough NOT to use are the female "atashi" instead of generic "watashi", for "I" and using the feminine "wa" to end sentences...
Then, of course, you have anime set in other time periods, or in "country-side" areas, which the writers like to do so they can emphasize certain "non-standard" accents or patterns of speech, such as the Osaka dialect, or Edo period Japanese, or similar. Trying to replicate that form of speech will definitely raise eyebrows.
Edit: an English example of this is provided by the most recent seasons of "The Walking Dead". Set in America, the vast majority of characters speak natural American English, but several characters have very quirky speech styles. The King uses florid, prose-filled English that everyone can understand, but no other person would use normally. The Garbage people use clipped English that is understandable but not natural at all. The character Eugene speaks with both a southern drawl and pseudo-intellectual vocabulary, oddly phrased. If people tell you that "Japanese in anime is different from real world Japanese" it is likely that they are referring to anime examples similar to these.