First of all Oblom is a puzzle game. It is designed to be hard, so that you can get those gray cells working.
The game is about a monster named Oblom who needs to reach the exit but there are boxes in the way. You need to move those boxes out of the way so that he can reach the exit.
1) Your goal is to clear the central path so that Oblom can go straight to the exit. The path is blocked by boxes.
2) There are 3 main operations -
You can switch the positions of 2 rows or two columns of boxes. You do this by clicking the left mouse above a column or to the right of a row and then clicking the right mouse at the column/row you want to switch it with.
Shifting the entire set of boxes
You can move all the boxes up/down/right/left by clicking that arrow key.
You have the chance of blasting 1 box by clicking on it. This can be done only once so use this option only when you really need it. A bonus is that when you blast one box all the boxes around it will also blast. So, this trick is useful but should be used only at the right moment.
3) If you did something wrong you can reset the whole map by pressing the 'r' key. This will put all the boxes in their original places as they were when you started
4)This is a puzzle game. The levels are generated randomly by the computer. You need to think of how to clear the central path of boxes ( using the above 3 operations ) so that Oblom go to the exit.
Tutorial video below demonstrates these concepts -