|Hey, she's his sister. I'm sure he can trust her.|
The speculation falls into two categories: The state of his health and the state of his government. Due to his weight and lifestyle, many have speculated that he suffers from related chronic conditions from gout to diabetes. There has also been some indications that he recently required surgery on a leg or ankle. Those seem quite plausible, but none of them should prevent him from making a public appearance if he wants to - however stage-managed it might be.
The other concern is some type or level of coup or change in practical or official leadership. Several names are routinely mentioned. Hwang Pyong-so is a General that has risen through the ranks of Kim's advisers and, primarily based on his leadership of the North Korean delegation to the closing ceremonies of the 2014 Asian games, has been seen as either a top adviser or perhaps a direct challenger to Kim and the leadership. Interestingly, there is also talk of the increasing power and importance of Kim's younger sister, Kim Yo-jong. She appears to be a truly trusted adviser, and speculation has run the gamut, from her running the government during Kim's medical recovery to her replacing him as a viable dynastic successor, as she is the daughter of Kim Jong-il.
The greatest challenge to a military coup is the longstanding cult of personality built up around the Kim family. Five generations have been fed a steady, endless diet of the greatness of their leaders and the necessity of having them running the state - surrounded as they are, on all sides, by existential threats. It would be one thing for the military to take power - they are by far the most powerful and wealthy organization in North Korea, and the troops under their command know the costs of hesitating to follow orders. So once they took power, the challenge would be to present a figurehead that wasn't a member of the family that could exercise power with credibility. Working to their advantage would be the fact that the people have been indoctrinated to never question the edicts from the capital. And in a perfect world, if Kim were to die - without ever re-appearing in public - a grieving Kim Yo-jong could assume the reigns of power - tightly controlled by the Generals and party leadership - as the direct descendant of Dear Leader I.
Why does it matter?
There are two issues around North Korean leadership - one immediate, and one further down the road. The immediate issue is, not to put too fine a point on it, war. Tensions are fairly high, with a recent exchange of fire across the border, and despite some indications of some openness to negotiations, no one is sure which direction relations are going. In addition, no one knows who wants what - it could be that Kim wants to improve relations with the west and the Generals don't, or it could equally be that Kim wants to order more provocations and the Generals are afraid he's going to start a war that will destroy them.
The future concern is the stability of North Korea as a nation. It's an economic basket case, with insufficient agriculture, no real manufacturing capacity, no international trade and no consumer base. Massively militarized, deeply corrupt and dependent upon China for basic viability, North Korea is one destabilizing shock away from utter collapse. And collapse would be chaotic and ugly in the extreme. The Generals and Party leaders would grab the hard currency and liquid assets and run for it. There are also nuclear weapons and weapons grade fissile material that would be impossible to secure. Millions of refugees would flood across the border into Southern China and northern South Korea - bad anytime, but horrific if it happened in a harsh Northeast Asian winter. And just as the reunification of Germany destroyed the German economy for a decade, the costs of normalizing and stabilizing a suddenly unified Korea would be staggering. Not even to mention that China would have to be a partner, and the last thing they want is a prosperous, western-allied Korea on their southern border. It's hard to guess what kind of Frankenstein's monster nation will emerge from that inevitable collapse.