The idea in the movie was that there is a microorganism called the Blight which is affecting the crops, one by one, to the extent that corn was the last crop that hadn't been significantly impacted.
It is indicated in the movie that the Blight breathes Nitrogen, which is registered as significant (it isn't, per se, but for the purposes of the movie, we take that as a given). Given that Nitrogen is 78% of the atmosphere, and Oxygen is only 21%, the movie implies that we cannot win. I think if you have to hang the entire "why we have to go to space", it's on that idea, take it or leave it.
I don't accept the specific reason, the Nitrogen-breathing, but I can imagine a microorganism that significantly affected crops across the board, to the extent that it's not a matter of extinction, but that there's going to be a very sharp rise in the price of food for a long time until the population declines into a much smaller equilibrium, e.g., human population before the advent of agriculture. The Potato Famine, which they mention in the film, is that on a smaller scale.
I think I would have reengineered the script with a line about "since agriculture adopted all of these crops with the common genetic modification that the Blight attacks..." But supply and demand curves (and politics especially) don't make for very good sci-fi.