To my (limited - I'm not a C developer normally) understanding, this is rooted in C. Remember that C does not know what classes or namespaces are, it's just one long program. Also, functions have to be declared before you use them.
For example, the following should give a compiler error:
void SomeFunction() {
SomeOtherFunction();
}
void SomeOtherFunction() {
printf("What?");
}
The error should be that "SomeOtherFunction is not declared" because you call it before it's declaration. One way of fixing this is by moving SomeOtherFunction above SomeFunction. Another approach is to declare the functions signature first:
void SomeOtherFunction();
void SomeFunction() {
SomeOtherFunction();
}
void SomeOtherFunction() {
printf("What?");
}
This lets the compiler know: Look somewhere in the code, there is a function called SomeOtherFunction that returns void and does not take any parameters. So if you encouter code that tries to call SomeOtherFunction, do not panic and instead go looking for it.
Now, imagine you have SomeFunction and SomeOtherFunction in two different .c files. You then have to #include "SomeOther.c" in Some.c. Now, add some "private" functions to SomeOther.c. As C does not know private functions, that function would be available in Some.c as well.
This is where .h Files come in: They specify all the functions (and variables) that you want to 'Export' from a .c file that can be accessed in other .c files. That way, you gain something like a Public/Private scope. Also, you can give this .h file to other people without having to share your source code - .h files work against compiled .lib files as well.
So the main reason is really for convenience, for source code protection and to have a bit of decoupling between the parts of your application.
That was C though. C++ introduced Classes and private/public modifiers, so while you could still ask if they are needed, C++ AFAIK still requires declaration of functions before using them. Also, many C++ Developers are or were C devleopers as well and took over their concepts and habits to C++ - why change what isn't broken?