What are the potential benefits of using inclined push-up bars vs the flat push-up bars?
What is the correct way of holding and exercising with this kind of bars?
Inclined bars allow you to shift your center of gravity, by altering the angle of your body to the floor. A steeper angle moves your center of gravity towards your feet, essentially shifting the effort required to do a pushup in that direction, so there's less strain on the upper body. Move down the bars and you increase the upper body effort required to do a push-up.
The surface of the push up bars are inclined at a slight angle to provide comfort to your wrist when performing push ups. Traditional push up bars have a flat surface which forces users to twist their wrist & lean their body forward. In Addition, the inclined surface of the new model helps you work your entire chest muscle whereas the old model only works your upper chest. The inclined surface also allows your head and neck to go lower, increasing the effectiveness of every push up you perform.
Basically, unless you have your feet elevated, they are arguing that you have to have your hands and arms at an angle at the top of the push-up, assuming that your hands wind up under your shoulders at the bottom. This tends to be a bit more pronounced when using bars like this because of how your grip works. If you're doing pushups on flat hands, your wrists naturally orient to bear the pressure, but gripping a bar means your wrists have to swivel sideways to hold you up.
Honestly, though, it's largely marketing. The actual purpose behind push-up bars like this is that it allows you that extra inch or two at your lowest point without bumping your stomach or nose into the ground. The angle is a minimal issue since moving your center of gravity forward a bit lets you operate with more or less a straight up and down motion. Changing the angle of push-up will shift some of the effort between the feet and the chest (as an elementary example, standing by a wall and doing push-ups off of it shifts the effort almost entirely to the feet. Doing pushups in a handstand against the wall shifts the effort almost entirely to the upper body). It will also exercise your pectoral muscles slightly differently, isolating the top, middle, and bottom parts.