The units digit is the ones place. In 12345, the 5 is in the ones place.
If we're looking at only the units digit, we can ignore the tens place, hundreds place and so on.
So 23 * 27 * 39 is like 3 * 7 * 9
3 * 7 ends in 1
1 * 9 ends in 9
Answer: 9
Note: The problem could have been 542578349023 * 5975348925379027 * 542903578349539 and we would still only be concerned with the numbers in the units digit (aka the ones place): 3 * 7 * 9. The answer would stay the same.