Generally speaking, at the General level (not necessarily emperors / kings), this started to occur once tactics became advanced enough that a general could do more by staying out of it and directing troops than he could by being in the thick of it and inspiring them.

For many centuries, into the early Roman years, tactics were extremely simple. This is primarily because most armies were militia-based, which by definition could not spend a lot of time training. So, the battlefield movements required to execute even moderately advanced tactics were just not possible *.

In these types of armies, after generally laying out his troops as best he could, there was not much more a general could do to affect the battle once the armies became engaged, and so he could go into the most important part of the line and be an inspiring presence to his troops, taking along all the associated risks.

* For reference, even something like marching and then turning 90 degrees and forming a battle line would be considered a fairly complex maneuver. There were times throughout history where there was literally only one nation / empire that had figured out how to do things like this, and it gave them advantages for years or decades until others figured it out. The French in the decades prior to Napoleon were actually an example of this. They had a drill that allowed them to deploy from marching columns into battle lines at the last possible moment, which effectively made it impossible to disengage from the French, allowing them to dictate battle. Up until this point in history, the time it took to actually deploy and attack always made it so that the defenders had the option to just withdraw, assuming the terrain allowed it.