Per the below order, I need to stay with the initial ATC frequency during an emergency, what is the benefit to do so? What if the aircraft entered into the airspace controlled by anther ATC or the aircraft is no longer within line of sight?
FAA Order J7110.65V, §2, ¶10-2-2:
Although 121.5 MHz and 243.0 MHz are emergency frequencies, it might be best to keep the aircraft on the initial contact frequency. Change frequencies only when there is a valid reason.
This doesn't mean you won't change frequencies if you move into another controller's sector or there's another reason for a handoff, such as moving from Center to Approach or Approach to Tower; it's to say that if you're already talking to ATC at all (often optional for VFR), then you should report your emergency there rather than switching to 121.5.
121.5 is monitored at every ATC facility, so if you have an emergency and don't know the right frequency to use, you can always go there and someone will answer you--and presumably give you a better frequency as soon as they've figured out where you are. If you already know the right frequency, though, then it's better for everyone (including you) to use that instead.
A pilot having an emergency is told to aviate, navigate, and communicate. Presumably the pilot has his/her hands full, and an unwarranted frequency change adds to the complexity.
If you were flying through my sector at FL370 and had to descend, I'd assign you FL350 and switch you to the next controller, who would assign FL240 and switch you to the next controller, who would assign you 110 and switch you to approach control.
If you were an emergency, I would likely coordinate with the underlying sectors, assign you 110, and switch you to approach control. Eliminating two frequency changes gives you the pilot more time to work the problem, hopefully leading to a successful outcome.
Here is another possibility, almost the inverse of @StephenS's suggestion. (Note that Stephen's suggestion is very much correct as well; if you are already in contact with ATC on a "normal" frequency, there is no reason to switch to 121.5 for the purpose of declaring emergency.)
Perhaps the paragraph should be interpreted like this:
Although 121.5 MHz and 243.0 MHz are emergency frequencies [and therefore should not be used except for initial contact, so as to keep the frequency clear for other potential emergencies], it might be best to keep the aircraft on the initial contact frequency. Change frequencies [away from 121.5/243.0] only when there is a valid reason.
In other words, the paragraph is telling the controller that the pilot remaining on 121.5/243.0 could be a safer option than having them change to the controller's normal non-emergency frequency. This could be the case, for example, if a VFR pilot is in IMC; a non-IFR-rated pilot will have limited experience flying without reference to the horizon and having the pilot move their head around—which would be necessary to adjust the radio frequency—could lead to disorientation and a loss of control.
As abelenky said in a comment, if you're talking to one controller, you already know you're in contact with them. If you call a new controller however, you have to establish contact, and they have to positively ID you.
I'm still a ATPL student and this is how i have understood from Airlaw. ATC will know on their display if you're out of their sight or not (they'll do all they can to not give you more pressure), and they'll make direct call to your next destination ATC when you are in a distress. Today we also have Data link system and the next ATC can also recive it on their screen and alert to the Rescue team in advance. And if you are out of sight you just need to switch frequency to your next ATC or the frequency you are on could still help ypu out or if you can switching to your next ATC and mention your Flight number exm: Birminghead-125 distress "postion" and "altitude" and your currently "situation" to update if the situation has escalated or not. That's how they do all the time, keep updating the situation is the key, and also aircraft around you sight can also forward your message etc.
Edit: i forgot to explain your scenario about benefits with not changing frequency is, if you have COM problems or COM equipment in aircraft doesn't work properly, or when you are transferring in blind so you don't need to switch to anyone.
