I phrased my comment badly. The word "really" absolutely can mean "for real", yes.
In this case, though, it mean's "very much so". If someone says "I don't like apples", we don't doubt that they dislike apples for real. If they say "I really don't like apples", it's still for real, it's just that theirs is an extreme dislike.
That would emphasize the intensity of his dislike of apples, and not the certainty of the speaker. The above sentence could be paraphrased as I am certain that he does not like apples. It does not say anything about the degree to which he does not like them.
If the sentence was On naprawdę nie lubi jabłek, then really would be a perfectly acceptable translation.