The red crosses indicate the moves which are available to you. If you think they disturb your concentration, uncheck the View moves toggle under the Options menu - the crosses are optional.
To play a move, just point at the square where you want to place your disc and left-click. If the move is feasible, a disc will be placed on the square and the board position updated.
Unless you have changed the settings, Zebra will promptly respond with a move of its own. The last move of the game is by default indicated by a small blue circle in the upper left corner of the square - this can also be changed in the Options menu by unchecking the Mark last move toggle.
For new players, the following feature is useful: To see the effect of a move without actually making it on the board, point at a square and press right mouse button and keep it pressed. This way WZebra will give you a sneak preview of the effects of the move. Once you release the button, the board position is restored.
As the goal of the game is to have more discs than the opponent, the concept of stable discs is important. A disc is stable if it can never become flipped by any move sequence from the current position. To see the currently stable discs, click the yellow/red icon. You will soon discover that in most games, very few discs are stable until only a few moves remain. This is one reason why having many discs early in the game can be bad for you; unless they are stable, chances are that the opponent will be able to flip them later.
If WZebra gives you a hard time, you might want to check out the Strategy guide for a brief introduction to Othello strategy.