What's the mechanical point of melee weapons' range?

The primary use of range is that it provides a constraint in the fiction of the game.

Dungeon World, Chapter 18: Equipment

Weapons have tags to indicate the range at which they are useful. Dungeon World doesn’t inflict penalties or grant bonuses for “optimal range” or the like, but if your weapon says Hand and an enemy is ten yards away, a player would have a hard time justifying using that weapon against him.

Hand: It’s useful for attacking something within your reach, no further.
Close: It’s useful for attacking something at arm’s reach plus a foot or two.
Reach: It’s useful for attacking something that’s several feet away—maybe as far as ten.
Near: It’s useful for attacking if you can see the whites of their eyes.
Far: It’s useful for attacking something in shouting distance.

It's just a cue for the discussion that ensues when opponents are using different types of melee weapons. For example, it may be harder to trigger the Hack & Slash move when your opponent is wielding a halberd[reach] and all you have is a dagger[hand]. In such a case, the dagger wielder probably has to do something interesting to get inside the halberdier's guard (Defy Danger, anyone?) and when they do, it becomes the halberdier's problem as they struggle to reel the sharp end in to strike the nimble dagger wielder. Hope you brought a backup weapon.

Another typical case is the environment, as the said halberdier may have a hard time within the confines of a castle or a dense forest, just to make up for their presumed superiority in an open field.

So when someone is trying to Hack & Slash, feel free to ask for more justification, and trigger other moves before (and if) you let it happen (at all).

You could well do all of this without the tags. The tags are there just to give it a bit of semi-official legitimacy, so that rules lawyers and sticklers have one less excuse to drag it out. ;)

Yet another use of the weapon range tags I've found within my group is to indicate whether a character is within attacking distance of something. My example is when character A is under attack by a monster (in melee) and player B is a few feet away. If character B wants to join in and counter-attack the monster, having a halberd makes this much easier than, say, a dagger.

In this case, character B might simply be able to deal her damage to the monster because it wouldn't have time to react to a quick thrust of a halberd while trading blows with character A. If character B had been using a dagger instead, the GM might have stated the monster noticed the rush toward the two of them and backed up enough to defend itself against them both, making the action trigger Hack-and-Slash instead. Or, knowing that it would be a (potentially risky) Hack-and-Slash situation, character B might only threaten to rush into combat range with the monster and simply trigger Aid for character A's next roll.

Additionally, standing in defense of an area would be simpler with a larger weapon (Close range for a battle axe perhaps) than, say, a shiv. The GM might be able to make a soft move to have the monsters move just beyond the reach of the defender and, while taking more time, still begin to pass by the hapless character.