Mostly this is just an explanatory errata, and in most situations it does not change the rules or common builds for barbarians monks and brawlers, which wished to make use of unarmed combat.
However, it is a significant change in a minority of use cases, where an ability focused on a weapon, rather than on an attack of some kind. This is explicitly explained by Jeremy Crawford when he says in twitter:
@DnDJester A monk can still use Stunning Strike with an unarmed strike. @pukunui81
For example. The spell to make a mundane weapon into a magic weapon, or to give a weapon some new property, does not apply to limbs or heads of creatures, no matter how well trained. Similarly, anything that requires you hold a weapon in your hand, does not apply to your hands themselves.
At first glance, one might think that this changes the rules for all Melee weapon attacks, since the "unarmed strike" was never intended to be listed as a weapon.
Weapons (p. 149). Unarmed strike doesn't belong on the weapon table.
However, in the Combat section, it explains that in any situation where there is a "melee weapon attack", the weapon in that attack can be replaced with an "unarmed strike". Even though it is not a weapon itself, it still can be used within a "melee weapon attack". This is an exception to the general rule regarding melee weapon attacks, which would normally require a weapon.
Melee Attacks (p. 195). The rule on unarmed strikes should read as follows: "Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons). On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier. You are proficient with your unarmed strikes."
This emphasis is based on the tweet by Jeremy Crawford who wrote:
Addressing a nuance in the PH errata: the rule lets melee weapon attacks use unarmed strikes, despite those strikes not being weapons.
When one is unarmed, they are by definition without a weapon, and a strike is therefore a melee attack, which is done without a weapon. One way to think about this is that all melee attacks fall into one of two categories, either a spell attack or a weapon attack, and since striking while unarmed is not a spell, it counts as a "melee weapon attack". Therefore most multiclassing builds or brawler builds relying on using unarmed strikes to trigger other abilities generally work. They are not intended to work however, with the feats that target weapons. For example, the Savage attacker feat. (reroll weapon damage dice)
Game designer Jeremy Crawford pointed out this blog post as a good summary of the rules change.
During this "transition phase" where the Monster Manual errata has not yet been released, Jeremy Crawford, also clarified that monster resistances are based on the damage type, and not if its a weapon or not, and monsters such as werewolves which are resistant to nonmagical weapons, should instead be resistant to "nonmagical weapon attacks."