That line makes no sense, that's why you don't understand it. As you said, it's casting to int* and then back to char*. It could just as well be written p = p; (or just left out altogether because obviously that's a noop).
Line 7 should be p = (char*) arr;. You need a cast there because only void pointers can be implicitly converted to other pointer types - other pointers or arrays can't (though some (most?) compilers allow it with a warning).
The cast on line 10 doesn't make much sense either. Since p has type char*, this line relies on the int pointer being implicitly converted back to char*, which, as I said regarding line 7, is illegal. And if it was illegal it would be the same kind of pointless to-int*-and-back casting we see on line 8. I.e. it calculates p+1 (using the type char*) and then casts that to int* and then right back to char*. So line 10 is the same as p = p + 1; (except the latter does not rely on an illegal implicit cast).
PS: Your main function is missing a return statement.
I am working creating a fully encapsulated, homogeneous singly linked data structure. The Listing class and SinglyLinkedList class that are part of the whole application compile fine, but the problem ...