An implicitly declared function is one that has neither a prototype nor a definition, but is called somewhere in the code. Because of that, the compiler cannot verify that this is the intended usage of the function (whether the count and the type of the arguments match). Resolving the references to it is done after compilation, at link-time (as with all other global symbols), so technically it is not a problem to skip the prototype.
It is assumed that the programmer knows what he is doing and this is the premise under which the formal contract of providing a prototype is omitted.
Nasty bugs can happen if calling the function with arguments of a wrong type or count. The most likely manifestation of this is a corruption of the stack.
Nowadays this feature might seem as an obscure oddity, but in the old days it was a way to reduce the number of header files included, hence faster compilation.