Vehementi (or whoever's in this photo) may benefit from a keyboard tray if he starts to develop a sore neck.

His torso is leaning backward, but his mouse hand is reaching forward. They're going opposite directions. People often do this when they're using a mouse that is not at the right height to use comfortably for an extended period of time. Leaning back allows you to shift the angle and height of your shoulder so you can still control the mouse without as much muscular tension in your arm, neck, and shoulder and without as much pressure on your wrist.
Is that bad? It may be. It shifts some of the strain to your low back and to a different part of your neck and shoulder. If this isn't his typical posture at the desk for 8+ hours a day, he'll probably be fine.
On the other hand now that I look at the photo some more, he looks like he's fairly tall and taller than the 5'8" person that most desks are designed for and putting the keyboard on top of the desk would proably place it at the ideal height for him. Most desks are designed to be optimal surfaces for the task of writing, which uses a different posture than keyboarding so the ideal writing height for someone who's 5'8" to 5'10" is around the ideal keyboard height for someone who's 6'2" - 6'4".
If that's the case then, he might be reaching forward because he's slouching and not the other way around like I originally thought.
Anyways... what it all means... it's all relative. There's nothing magical about a keyboard tray, but they tend to be helpful for various reasons. If all you do is keyboard on that desk, the tray is just another surface. If you need to write and keyboard a lot, a tray is usually a good idea.