What is C?
It's one of the very old programming languages still around, as it was first free in 1973. Because it's old, it's become well-established in a wide variety of programs. Operating systems and device drivers are usually written in C, because the language is low-level and depends on only a few things, which is an essential obligation for operating systems. It can as well be optimized to run very rapidly, another important prerequisite.
So why should you study it today, if you aren't involved in operating systems or device drivers? One significant reason is that the language is very powerful. Modern languages for example C++ and Java have big similarities with C language, and learning C will make it easier to study those languages in the future, as they share much of very similar syntax and general design characteristics.
As well, C forces you to think about things that additional languages would gloss over. This also applies for memory management. If you require memory dynamically, you have to by hand allocate and free it on the heap, or else you run into problems with memory leaks. There are no classes, so you must describe structs and unions, two different datatypes that you can make use of to hold more than one variable at once without the help of object-oriented programming. Once more, this forces you to think at the low-level and in fact understand computer science concepts instead of just smashing code blocks mutually.
It's certainly true that for many programs, using C causes more problems than it solves. For anything that you aren't doing on the low-level, it's generally easier to use a higher-level language and its functions in place of trying to do everything by hand. On the other hand, even if you don't use C in the future, your programming in other languages will get better because you know the underlying standards of the tools and libraries that you are using, so you can then make use of the correct ones for the job in place of guessing.
Conclusion is, C's simplicity is what makes it so influential of a language. It's practically a blank slate, which creates it the ideal way to study computer science concepts and well-organized programming.