There certainly are reasons to see this movie, but they won't make this a great movie. It's a good movie, though, and good fun. Based on a very true story, the rise of this very bad singer into popular consciousness is worth telling.
As always, Streep is virtuosic. Maybe that's a problem in a way, because they depend so much on her being convincing and compelling that they use too much of her. Impossible? Not at all—the plot is simple enouch as it is, and it stalls with long sections of Streep singing really badly really well. It's not easy to be bad on purpose. But it you itch for more sometimes.
Hugh Grant? An interesting and not quite canny casting move (is this a problem with having more than one casting director?). He's stretching himself slightly out of the typecasting he's been stuck in (successfully) for years. And it actually clicks pretty well, and yet isn't quite a fit for Streep. Or for the time period (WWII New York, more or less).
You can read the remarkable details of this woman's life elsewhere—and in the final credits—and it's what holds it all together, of course. But this is a movie, and it doesn't fill in and make vivid the experience (the movie experience) fully. You'll see. Enjoyable but ultimately thin.