What do you mean exactly by "software to redesign" it? Design is normally done with a combination of graphics software, word processing software, pen and paper, and perhaps a white-board. You might approach your task by creating a flow-chart detailing the player's progress through the available menus and gameplay screens, and create diagrams for how you would like to lay out each screen. It can often help to create a couple of imaginary players, and track how a typical gameplay session might go for those people. Being a relatively simple game you could easily create a playable prototype by cutting out and drawing bits and pieces on paper.
As for how that game is made: It's a Flash game. There are a number of approaches, but the common ones would be to use the official Flash software from Adobe (which your school might have?), or to use FlashDevelop with HaXe or ActionScript (which is probably more common).
What sort of experience do you already have? Do you already know any languages? Are there any requirements for what languages or tools can be used?
Do you need to re-create the game as a web-game, or can your version be for desktop (which may be simpler as a beginner)?
Do you have to code, or could you approach the problem using a simple tool such as Construct 2 or Game Maker (again, making things simpler for you at the expense of learning more about coding)?