You're right that some low-level knowledge doesn't matter. For example, I think you'd be nuts to routinely re-jigger arithmetic to replace multiplication and division with equivalent bit-shift operations to save a cycle or two. The speed-up is likely to be microscopic, the compiler might do it anyway, and I suspect doing things like triangle_area = (base >> 1) * height
is just asking for trouble when someone looks at your code in six months. Similarly, a lot older code jumps through incredible hoops to reduce the number of floating point operations; that's also not worth the hassle, mental overhead, and potential for bugs these days.
On the other hand, there are other low-level concepts that can give you relatively large gains for minimal work. For example, consider caching. For example, you could iterate over an NxN matrix like this:
for(i=0; i<N; ++i)
for(j=0; j<N; ++j)
do_something(matrix[i][j]);
or like this:
for(j=0; j<N; ++j)
for(i=0; i<N; ++i)
do_something(matrix[i][j]);
If the matrix is suitably large, one of these (depending on how the language lays out the array) will cause a cache-miss for every single array access, while the other will be optimal. Assuming you've got to iterate over the matrix somehow, why not prefer the potentially faster one? They're equally readable, equally writable, etc.
In general, I think appreciating the things going on "under the hood" can help inform your decisions when programming in higher-level languages (e.g., Can I code this in a way that doesn't trash the cache over and over again?). If you're unfamiliar with the relationship between an array index and a memory address, you might not appreciate how important it is to check array indices and buffer sizes. I think you could glean a lot of this information from the higher-level language's documentation, but it'll definitely be more in-your-face in C.
Obviously, the trick is not to go overboard. There is a theoretical speed difference between i++ and ++i, but if copying a single four-byte variable makes or breaks your web-app, the app is already in pretty dire straights.