You can damage yourself, if you really like
The cone's point of origin is the square the caster is in, or if you're using more abstract ways of measuring, the point of origin is the caster themselves. (Their hands, their mouth, whatever the spell's flavour decides.)
Spells that create [areas of effect] that originate from you also have a range of self, indicating that the origin point of the spell's effect must be you. (PHB p.202, emphasis mine.)
This rule allows you to include the origin of the cone (yourself) in the effect. This most likely doesn't ever really seem like a good idea unless there's a Swarm in your space, but in that case, you are allowed to include your own space in the effect, else you'd never be able to hit something inside your own space with a cone spell.