Body weight programmes can be very effective for gaining muscle and losing fat. For example take a look at gymnasts, martial artists, and free runners.
The programme you posted looks a bit shit.
If main goal is to lose weight you should look at the food you are eating and see if there are some small changes you can make, don't go overboard and cut all sugar carbs and fats all at once, just little incremental changes over a longer period of time.
Stick to the basics if you have no gear
Push ups, dips, chin ups, box jumps, squat jumps, burpees and sit ups.
Id suggest you invest in a barbell and some plates, look around online for a cheap second hand set. Even without a bench or rack you can perform a lot of fundamental exercises with a basic bar and a couple of plates
If you are DIY inclined have a look on the interweb for home made exercise gear it could be something as simple as filling up some 3ltr milk bottles with sand and putting them in a backpack for extra weight for chins and press ups.