I don't think that's exactly what that says though, the full quote is: "“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.”
Depending on how you parse that, it could be saying that using a weapon isn't the only way to make a melee weapon attack, you can also use an unarmed strike to do so.
Further, Crawford clarified in this tweet thread:
http://www.sageadvice.eu/2015/09/06/what-specifically-does-melee-weapon-attack-mean/
It seems that Unarmed Strikes, while not weapons, can be used to make a melee weapon attack. Personally, I think all of the push for "plain English" that they were going for is now biting them in the ass.
Melee Weapon Attacks (an attack with a melee weapon or unarmed strike where the weapon stays in your hand), Attacks with Melee Weapons (which include throwing melee weapons but not unarmed strikes), etc...
ETA: But yeah, based on the spell description, I don't think it affects unarmed strikes which are decidedly not weapons. With that said, who knows if the spell was written with that paradigm in mind. I think it came out before the PHB errata.
