Explaining the Value of Knowledge of Digital Logic
Your best bet of how to answer why students should care is to stop and ask yourself what good reasons there are to care about it (or any topic). Do that before reading on?

OK, so my answer is: because science. The word means knowledge. If you want to know something instead of just have opinions (possibly wrong) or worse, then you will want to be finding out. But why bother to know anything, is really the question. Caring is your issue. Caring means that one can influence things, and not caring means that one will be pushed around helplessly by circumstances.

Is it effective to care, is the obvious question? Can we actually learn enough to have any effect? When things seem perplexing and obscenely complicated, then it appears hopeless. Science brings hope because it is able to render the complex simple enough to manage. When I found out that all matter is composed of just 3 elementary particles, and that there are basically only 2 forces that affect us, only one of which we have any control over, it was revolutionary. If more people knew these things, they might wish to learn more. When you ponder the 88 keys on a piano, able to make basically all (western) music, or the 92 naturally occurring elements making basically all things, it is astonishing. The organization of atoms (predicted 2500 years ago) in to the stunning wonder of the Periodic Table is a great story!

All computers are made of transistors, and built up in to a few simple circuits replicated millions of times, like the cells of the body or brain. If that is not inspiring and provoking of interest, I would release the students from school, as they are unreachable: they don't care to have an influence on their own future.