The great idea about GANs is that they replace one of the most hard to understand parts of a neural net - the "Loss function" - with another neural net, thus making the loss function learnable. This opens up the door for a kind of unsupervised learning that was impossible to make work before. GANs are very very important also because they are almost like reinforcement learning (actor + critic = RL, generator + discriminator = GAN), and RL is supposed to be the way to AGI.
The most famous problem of GANs is instability during training and mode collapse - which is like a student learning especially for an exam (and not in general) thus optimising for the test instead of the real thing.