why cant I print out the numbers by writing *p[3] instead of just p[3]
The expression p[3] is, by definition of the subscript operator [], equivalent to *(p+3), which means that the int element that exists 3 elements after the element pointed to by p is retrieved.
Therefore, *p[3] is equivalent to **(p+3), which does not make sense, because *(p+3) is an object of type int, which cannot be dereferenced. Only pointers can be dereferenced.
And why isn't it int *p = &array instead of int *p = array?
In the declaration int *p = array;, the expression array will decay to a pointer to the first element of the array, i.e. to &array[0]. Therefore, p will point to the first element of the array.
However, if you write int *p = &array; instead, then array will not decay to &array[0]. Instead, the expression &array will evaluate to a pointer to the entire array, not to a pointer to the first element of the array. This means that the type of the pointer is not int * (pointer to a single int), but rather int (*)[5] (pointer to an array of 5 int elements). A pointer of type int (*)[5] cannot be assigned to a pointer of type int * without an explicit cast. For this reason, the line int *p = &array; is not valid.