The 159 constraint is the most common form of “Column Indexer” constraint.
In each row, the digit in column 1 indicates the column in which the digit 1 appears. For example, if column 1 contains a 4, then column 4 is 1. If column 1 is 7, then column 7 is 1.
A corollary is that if column 1 indicates where digit 1 appears, then the column in which digit 1 appears indicates what is in column 1. For example, if column 4 contains a 1, then column 1 is 4. If column 7 is 1, then column 1 is 7.
The same is true for every digit in columns 5 and 9.
Some puzzles tagged with 159 may include additional indexed columns, or indexed rows.