Nikoli describes the rules in this way:
1) Fill in all empty cells with numbers under the following rules.
2) Divide all of the board into blocks. Fill each block with the same number horizontally or vertically.
3) Each block contains as many cells as the number in the block.
4) Same sized blocks cannot touch each other, horizontally or vertically.
I'll walk you through this small example puzzle.
A good place to start is with numbers that are blocked in by other numbers and the edge of the grid. The 2 and the 3 in the top right of the grid need to be part of two 2s and three 3s respectively as part of the rules. Those groups can only go in one direction, like this.
Another thing to watch out for is ending up with too many of a particular number. The 2 in the bottom left can't extend to the right because then there would be a group of three 2s,a nd there has to be exactly two 2s. So it must go up.
What would happen if the 2 in the bottom right extended upwards?
The remaining area in the bottom right of the grid couldn't work. It can't have any more 2s because they would attach to the set of 2s we just placed. It can't contain anything bigger than 2 because there's not enough space. It can't contain a 1 because then there would be a square left over. So it doesn't extend upwards. You can make the same argument in the top left of the grid.
We're almost done, and the only incomplete clue is the set of 3s. Could it extend down?
No. The large space with area three can't contain 3s, or two 2s, so that area couldn't be filled.
We're left with this space and no clues. There can't be any 2s in there because all squares border existing 2 areas. Similarly there aren't three spaces that can take 3s, so it must be full of 4s.
