What is the significance of these lines in the poem:- "One can see what will trouble This sleep of mine, whatever sleep it is. Were he not gone,..."Also please state the meaning of these lines...
Most critics do agree with the first answer. However, I do not think that the poem must necessarily be about death. I think that it is quite possible to look at it more literally. Since you seem dissatisfied with the first answer, I'll try to discuss it in that sense.
Reading the poem literally, it is quite clear what will trouble his sleep -- he'll be seeing apples everytime he closes his eyes. I have had this same experience where I'll see fruit moving around in front of my eyes if I just spent all day picking it -- I close my eyes and it's like I'm still picking fruit. And he may be so tired that he has a hard time getting to sleep -- that can happen too.
In this interpretation, it's pretty hard to account for the woodchuck, though.
The lines pursue a rather Shakespearean idea of a relation between sleep and death. The idea is articulated in Macbeth where sleep is seen as 'death's counterfiet' or in the sonnets where Shakespeare calls sleep 'death's second sleep'. Sleep is temporary death whereas death is an irrevocable and infinite sleep.
The entire poem After Apple-Picking by Robert Frost may be seen as the dream-laced thoughts of an old dying man who is ready for death with a sense of completeness but like all other completenesses, this one is also mingled with the lingering trace of a desire for life.
Towards the end of the poem, the speaker wants to make a distinction between the sleep which has enraptured him for the time being and the eternal sleep that is in the offing. But in tune with the radical unknowability of death, the difference can never be known because the one who knows (the woodchuck) it is always already departed.