This comes up incredibly frequently, and having it be second nature will benefit you.
A lot of the time, this gives you a starting place for how your inputs and outputs should look, what your method signatures should be. For example, if your data can can fit in a uint8_t, why waste bytes on a uint32_t?
This isn't the be all and end all, especially since modern architectures use larger word sizes/RAM is plentiful/bandwidth is cheap, but it can be a helpful way to frame the conversation when you're architecting software (especially network protocols).
I'm an ME who went into embedded systems, so I'm not sure what qualifies as low level for a web developer and if checking timing using an oscilloscope or saleae is feasible for the type of work OP wants to do. But even just looking at the assembly would have made things obvious in my example.
EDIT- This made me realize that a rudimentary understanding of the assembly language for your architecture will be very valuable. Doesn't have to be enough to actually write any code in it, but it's great to be able to take a look at what was generated when things are acting weird.
Without that though, performance is really really bad. The inner loop of a motor controller is no place for them...
Anyway a bit toggle tells all, and if you're making speed optimizations you should be checking that things actually got faster.
So I agree, knowing how data is stored is critical.