A writer is someone for whom writing is more difficult than it is for other people.
Mann is describing not a gift but a peculiar affliction—the writer's curse is that language never quite surrenders to intention the way it does for others. Most people can dash off an email or explain themselves in conversation with reasonable success; the writer, by contrast, feels the gap between thought and words as an almost physical pain, revising endlessly because the first draft always betrays something essential. It's a useful corrective to the romantic notion of the inspired scribbler, reminding us that Flaubert spent weeks finding the right adjective not from pretension but from genuine torment. This also explains why the best writers often sound so careful on the page—they've earned that precision through suffering that casual speakers never experience.