Kakuro is a crossword where the answers are digits and the clues are sums. The version on zifrio is the open variant: the numbers along the edges tell you what each row and each column must add up to. You fill the white cells with digits from 1 to 9, and no digit may repeat within the same row or column.
How to solve it
Start at the extremes. A run of four cells can total no less than 10, because 1+2+3+4 is the smallest possible, and no more than 30 with 6+7+8+9. When a sum sits near either limit the options nearly vanish: a four-cell run totalling 11 must be 1, 2, 3 and 5 in some order. Runs like that are the anchors for the rest of the board.
Then work the crossings. Every cell belongs to both a row and a column, and its digit has to fit a valid combination for each. That is usually where the solution locks into place — a digit that works in the row but appears in none of the column's possible combinations can be struck out.
Keep track of what is left. Once three of four digits are placed in a run, the last one is plain subtraction.
What makes a puzzle hard
Grid size, mostly. Four by four fits in your head; six by six puts more cells behind each sum and therefore more combinations to sift. Sums in the middle of the range are also more awkward than extreme ones, precisely because they admit more sets of digits.
A worked example
A run of five cells totalling 17. The smallest possible is 1+2+3+4+5, which is 15, so you sit just two above the floor. The only ways to reach 17 with five distinct digits are to swap the five for a seven, or the four for a six. Two options instead of hundreds — and the crossing columns settle which one quickly.
The most common mistake
Forgetting that digits cannot repeat within a run. It is easy to find a set that hits the target and miss that it contains two fours. The check is simple: the right total is not enough, the digits must also be distinct.