When the ability says "innate spellcasting feature" this means things like Dragons breath attacks and Beholders ray attacks right or a spell via Magic Initiate feat right? If so and then take for example the Spell Thief stole a Dragon's Breath attack, does it mean the dragon has a to roll to recharge its breath attack right before it can use it again thus literally robbing it of its use temporarily?
No. This means exactly what it says. Many monsters have a feature called Innate Spellcasting. For example, the Storm Giant has:
Innate Spellcasting. The giant’s innate spellcasting ability is Charisma (spell save DC 17). It can innately cast the following spells, requiring no material components: