I'm wondering, Is it possible to gain muscle by doing this workout every day or should I workout with weights or other equipment to gain muscle?
As always with workouts, whether it is "possible to gain muscle" from it depends on where you start. If you lead a sedentary life and don't work out at all, then you can probably benefit a fair amount from this workout, but you won't end up looking like the guy in the video by doing it.
The video tells you to repeat the workout 2-3 times, which lands you at 40-60 minutes, which translates to ~4.5-7 hours per week if you do it every day. That's quite a lot of time, which if spent at traditional weight training would give far better results (if muscles and strength is your goal). Heck, a beginner with a good program (like Starting Strength) can spend half that time at the gym and still get quite good results!
I'd also like to point out that despite its advertising, this is not a full body workout. Most of the exercises included are core and/or cardio; there are two pushing exercises (burpees and pushups) which work chest and triceps, three leg extending exercises (burpees, squats and glute bridges) which work your thighs, but there's no pulling movement at all to work upper back and biceps. So to make this a true full-body workout you'd need to add something like pull-ups or inverted row.
And what are the benefits of doing this workout every day beside weight loss?
Well, a strong core is generally a Good Thing for most occasions in life, and cardio is good for your health. As for weight loss, exercise isn't very effective for losing weight, unless you also restrict how much you eat.
