This formula is evaluated for each of the 10 cells in A1:D10. A1 will change to the address of the cell being evaluated, while C1:C10 is entered as an absolute address, so it won't change at all.
The key to this formula is the =0 at the end, which "flips" the logic of the formula. For each value in A1:A10, COUNTIF returns the number of times the value appears in C1:C10. As long as the value appears at least once in C1:C10, COUNTIF will return a non-zero number and the formula will return FALSE.
But when a value is not found in C1:C10, the COUNTIF returns zero and, since 0 = 0, the formula will return TRUE and the conditional formatting will be applied.
Named ranges for simple syntax
If you name the list you are searching (C1:C10 in this case) with a named range, the formula is simpler to read and understand:
This works because named ranges are automatically absolute.
If you need a case sensitive count, you can use a formula like this:
The EXACT function performs a case-sensitive evaluation and SUMPRODUCT tallies the result. As with the COUNTIF, this formula will return when the result is zero. Because the test is case-sensitive, "apple" will show as missing even if "Apple" or "APPLE" appears in the second list. See this page for a more detailed explanation.