At the end of S08E06 of Game of Thrones, Jon Snow along with Tormund and others of the Night's Watch
leave the Night's Watch.
and go into the woods.
Where are they going? And why?
In S08E04 The Last of the Starks, when Jon is saying goodbye to Tormund, he tells Tormund that the freefolk are welcome to stay in Winterfell but Tormund tells him he will take his people beyond the Wall where they belong, where they like to wander around. He also says to Jon to come with him beyond the Wall as Tormund believes Jon has got the North in him. The real North.
Tormund: I am taking the Freefolk home. We've had enough of the South.
Jon: This is the North. Freefolk are welcome to stay.
Tormund: It isn't home. We need room to wander. I'll take them back through Castle Black as soon as winter storms pass. Back where we belong.
Jon: (looks at Ghost) It's where he belong too. A direwolf has no place in the South. Will you take him with you? He'll be happier up there.
Tormund: So would you.
Jon: I wish I was going with you. This is farewell then.
Tormund: (hugs Jon) You never know. You've got the North in you. The real North.
When Jon is sentenced to a life of serving the Night's Watch, he goes there and finds Tormund.
At this moment, the North is an independent kingdom with Sansa as the Queen. Jon is done serving his people, doing his duty. If he stays at Castle Black, he would have to be a Lord Commander possibly.
But he decides to just go away from his vows and duties and promise and just be free for once in his life. Free from all the burden of expectations of who he is supposed to be.
Edit 1:- based on comments
1. It is very unlikely that Jon is going beyond the Wall for ranging. The Night King and Army of the Dead are defeated. There are no threats to range for. Also, he was the only one from the Nights's Watch going with the Freefolk.
2. As suggested by @krb, Jon turns back one last time towards the Wall with a melancholic look, a bittersweet look if you will, towards the realm he guarded for so long.
Renfro: Did you have any conversations with showrunners David Benioff and D. B. Weiss about what is going on with Jon specifically? I feel like the shows ends on this optimistic note, where he's half-smiling and there are children around him. The children's presence felt really important in that scene, because it's this mark of the future and possibility. At the same time it's a little sad, because he's going into exile, basically, and leaving behind his past life. Were there conversations about that?
Djawadi: Yeah, absolutely. The idea is really that he stops and he looks back, and then the main title starts, and it's the idea of a new beginning. It's supposed to be positive and yeah, like you said, the fact that there are children around and [other] people — he's not just by himself.
Originally [he was] with the Night's Watch, and you're not allowed to have a wife and children and all that, but this is him going out there with the wildlings, and you can interpret it like he's starting a new life. He's a changed man, and he's leaving the past behind, and so it's definitely supposed to be something positive. There are many possibilities now — that's how we can look at it.
Without any dialogue or out-of-universe explanation from the writers themselves, the only objectively true interpretation is that the reason for Jon traveling north is left as intentionally ambiguous.
Is he accompanying the free-folk to guide them to a new home? Is he doing a favor for Tormund? Is he taking Tormunds invitation to stay with the free folk? Does he intend to stay indefinitely, or will he return to the wall? Does he even know at that point?
All questions we have no answers to, and due to the episode being the finale of the series it is unlikely we will get any definite answers apart from post-story expositions, which tend to be by nature half-baked.
Does Jon really abandon the Night's Watch, or is he still keeping his vows?
In the beginning of ASOIAF, the duty of the Night's Watch consisted of manning the Wall to protect Westeros against the wildlings. Later on, it shifted to protecting the humanity against the Night King and his armies. Now, the Night King is defeated and the wildlings are no longer enemies. Therefore, assisting them in reclaiming the lands behind the Wall amounts to strengthening the Northern border against any unknown enemy and is entirely within the duties of the Watch.
One more thing in favour of Jon staying a part of the Watch: the lords of now Six Kingdoms were happy to see Jon take the black once more because the vows include the oath to father no children. The last thing they need is the new generation of Targarien pretenders. He'd better not break this one, or they will come after him with a vengeance. Bronn's happily retired from the hitman career, but as the Master of Coin he'll hire a few Faceless Men even if it delays the vital task of rebuilding the brothels.
The time Jon spent among the free-folk was the happiest period of his life. He fell in love with Ygritte, but beyond that he had an affinity with the people and the way they lived without tyranny. This is all very well illustrated in the books, but the clearest evidence we have in the TV show is the foreshadowing conversation he has with Tyrion in the final episode..
"Love is the death of duty."
"Sometimes duty is the death of love."
He's not just referring to Danaerys, but his first true love, Ygritte, who died in the opposing army when the wildlings attacked Castle Black.
His heart always belonged with the free-folk. His sense of duty pulled him away, but now he is finally released from duty.
There is no indication that Jon has abandoned the Night's Watch.
Rangers of the Night's Watch journey North all the time for a variety of reasons.
I read an interpretation that when it's suggested that Jon take the black and re-join the Night's Watch as his sentence to appease Grey worm and the Unsullied, Grey Worm is unaware that the Night's Watch is now near enough non-existent (through all the men lost in battle, and the fact that there is no longer a pressing need for them).
This means that Jon is free to return to Castle Black having 'taken the black' and can meet Tormund before winter is over so that they can venture beyond the wall as free-folk.
I’d argue that the proof that he’s remaining with the Nights Watch is that he once again put on his black uniform, and wore it as he accompanied the wildlings north.
I’d expect that if he were deserting the watch he’d wear his own clothes rather than his uniform.
Edit: Jon in the last scene is actually a mimic of the first scene of season 1. Where people were running for their lives north the wall. Now its a safe home again, and he's the shield that protects the realm of men. You can look at popular posts online, the last scene a the first scene share a lot of the same shots.
Jon is clearly just escorting the wildlings north of the wall now that Spring is here. Its important to have someone go and physically know the location of their settlement. Jon is good friends with Tormund there's no reason for him not to go.
Tyrion and Sansa are expecting him to be at castle black, or somewhere at the wall. There's no evidence of him abandoning the nights watch. He would handed over the sword of the last lord commander or something of that nature. And I'll have to re-watch it of course, but just by the way he was walking around and the way some knights watch men were escorting him it seems like they've already accepted him as lord commander.