Backpropagation is a leaky abstraction; it is a credit assignment scheme with non-trivial consequences. If you try to ignore how it works under the hood because “TensorFlow automagically makes my networks learn”, you will not be ready to wrestle with the dangers it presents, and you will be much less effective at building and debugging neural networks.
That is from the excellent Andrej Karpathy, “Yes you should understand backprop”.
I say it’s possible to use deep neural networks quite effectively without truly understanding backprop. But if your goal is to specialize in the field and apply this tool to a range of problems, then “yes you should understand backprop”.
By the way, @karpathy is a prolific Twitter feed with 37,100 followers.