There are a myriad reasons why they're popular again.
1. Hardware has caught up, and is cheap. When Backprop was invented back in the 80s, you couldn't train networks with more than a couple of 1000 nodes tops. Today, with GPUs, you can train networks with billions of parameters.
2. More data is available. Back in those days, you had a few dozens (maybe a few 100s) of examples in your training set. Today, people play with sets larges than 1TB.
3. Dramatic successes. For a while, the ImageNet competition was seeing slow and stead progress. Then DL comes along, and there's a 20% jump in performance (I'm too lazy to look up the exact numbers...). If you've ever competed in such competitions, progress is painfully slow (see, for example, the Netflix competition). So a jump of that magnitude in performance in 1 step is mind-blowing. On top of that, every year since then, the performance has increased significantly.
These are just 3 that come to mind.