Each bullet point of mage slayer (PHB, pg. 168) is a separate ability:
- When a creature within 5 feet of you casts a spell, you can use your reaction to make a melee weapon attack against that creature.
- When you damage a creature that is concentrating on a spell, that creature has disadvantage on the saving throw it makes to maintain its concentration.
- You have advantage on saving throws against spells cast by creatures within 5 feet of you.
As long as you attack a creature who is concentrating on a spell, you can force disadvantage on their saving throw for concentration. The prior bullet point simply grants you reaction attack if a creature next to you casts a spell. Although if you hit, then they would have disadvantage on their concentration save per the second bullet point.