In S05E05, I didn't catch the reason why Jon Snow needs to establish an alliance with the Wildlings. Just to help Stannis in his war to the North? And if he has good reasons, why wouldn't the other watchmen understand them?
Note: I don't have the episode to refer to here for direct quotes, so my answer won't be directly sourced. I can try to go back and fill them later if needed.
As far as you're comment about Stannis, Jon isn't trying to help him with his war. He told Stannis plainly that the Night's Watch would take no part in any war and even personally refused Stannis's offer to leave the Night's Watch and join him in retaking Winterfell.
Jon's reasons are both practical and humanitarian
The practical reasons
The real reason he wants an alliance with the Wildlings can be summed up like this: The enemy of my enemy is my friend.
On their own, the Night's Watch has nowhere near enough men to hold back the White Walkers. A constant theme in both the books and the show is that the Night's Watch is severely under-manned and isn't being given the resources and manpower it needs to do its job. Most of the fortresses and castles along the wall are unmanned and in disrepair (you'll remember that the order Janos Slynt refused was to go repair and fortify one of these called Greyguard).
This situation was bad enough when the Night's Watch just had to deal with the Wildlings. But the White Walkers are such an overwhelming threat that there's just no way the Night's Watch can hope to contain it. So, to have any chance of survival, Jon needs allies. Since the rest of the Seven Kingdoms is mostly content to ignore the repeated warnings coming from the Wall, Jon turns to the only potential allies he has left: the Wildlings.
Jon also knows that if the Wildlings fall to the White Walkers, the only purpose it would serve would be to make the Walkers stronger. Every Wildling killed becomes one more member of the horde.
So, either he makes common cause with the Wildlings, which offers a small hope of defeating their common enemy. Or he leaves them to their fate, which only swells the ranks of the White Walker army.
The humanitarian reasons
Jon knows that the White Walkers represent an existential threat to all men everywhere, regardless of whether they live north of the Wall or not. He makes a comment that the Night's Watch is supposed be "the shield that guards the realms of men." He goes on to say that the Wildlings are men and so the oath applies to them too.
Why isn't the rest of the Watch on board?
It shouldn't be hard to guess. Most of the watch have lost friends and brothers to the Wildlings. Jon himself says he lost 50 brothers during their attack on the Wall. They have all spent their entire time in the Watch up to this point fighting against the Wildlings. That's a lot of history and baggage to overcome.
In the show, they present this most explicitly in the character of Olly, who serves as Jon's steward. He watched the Wildlings murder his parents and most of his village during a raid. He understandably hates the Wildlings for this. And now Jon wants to ally with them and give them land south of the wall to settle on. So, from Olly's perspective, they murdered his parents and are now being invited to move in next door.
He needs the Wildlings south of the Wall because every person north of the Wall is a potential wight, the foot soldiers of the White Walkers. Anyone who is killed can then be turned into a wight, boosting the size of the White Walkers' army. The bigger the army the White Walkers have, the more difficulty the Nights Watch will have in defending the Wall when the Walkers attack.
This is complicated by the fact that many of the Nights Watch don't believe Sam about the army of the dead and the White Walkers. They have not seen the threat the White Walkers pose, they have only seen the threat of the Wildlings, and so the Nights Watch is not a fan of providing the Wildlings with amnesty and free land.