I watched the series a few years ago and I can't really remember what happened. Does anybody remember if the reason for the existence/why the walkie talkie was working ever explained?

Thanks for answering in advance haha~ I won't be able to rewatch the series XD

No it was not explained but it seems like it's because they were both involved in the Inju case and how much of an impact it had on the two main characters. I'm not sure though

Everything before 2000 for Jaehan was an alternative Haeyeong, on the psychiatric hospital is the first contact with our current Haeyeong and as he helps Jaehan the former present is changed, all off their connections relied on Jaehan's death on August 3rd, 2000 at 11:23. Since the message that tipped Jaehan about the hospital was Haeyeong's last, the following message would have been Haeyeong's first, but the past was changed and no more radio transmission.

because of determination and will of detective Lee. He died at 11.23 pm that is why LT. park recieved signal at this time. Both 2000 and 1999 case were related to Both Park and Lee. 

When Detective was about to die , he want to change the past because all of his cases were never solved. He himself said " no case will never solve if I will die". 

And Detective possession was his walkie talkie at the time he died.