In the formula, COUNTIFS counts the number of times each value in a cell appears in its "parent" column. By definition, each value must appear at least once, so when the count > 1, the value must be a duplicate. The references are carefully locked so the formula will return true only when all 3 cells in a row appear more than once in their respective columns.
The helper column option "cheats" by combining all values in a row together in single cell using concatenation. Then COUNTIF simply counts the number of times this concatenated value appears in column D.
Helper column + concatenation
If you don't mind adding a helper column to your data, you can simplify the conditional formatting formula quite a bit. In a helper column, concatenate values from all columns. For example, add a formula in column E that looks like this:
Then use the following formula in the conditional formatting rule:
This is a much simpler rule, and you can hide the helper column if you like.
If you have a really large number of columns, you can use the TEXTJOIN function (Excel 2016 365) to perform concatenation using a range:
You can then use COUNTIF as above.
If you're using a version of Excel before 2007, you can use SUMPRODUCT like this: