'\0' is a character literal specified as an octal escape value. Its value is equal to 0.
The declaration
char inputName[80] = {'\0'};
means that the first element of the character array is explicitly set to zero that is to character literal '\0'. All other elements of the array also will be set to zero if there are no explicit initializers for other elements of the array. The same result you could get if you would write (only in C++)
char inputName[80] = {};
That is in the both cases all elements of the array will be initialized with 0.
@dritail
The program prompts the user for a file but you don't really know how many elements are in the file so why did they use 80? when declaring the variable "inputname"
They simply decided that 80 is a reasonable value for the file name.:)