Actually it means exactly that.
The way you put it ("supporting your right to say it doesn't mean you shouldn't suffer the consequences of your speech") is totally absurd.
"Your right to say it", means exactly the right to say it and NOT suffer consequences.
If we accept "suffering consequences" as orthogonal to "having a right to say it", then we come to the absurd conclusion that even people in the most opressive regime, or those threatened by the Inquisition had "free speech".
They said it, and then, they suffered the consequences (from getting fired to getting set on fire, depending on the period).
Free speech means anybody can say anything, and people supporting his right to say anything, without any personal consequences (other than others potentially disagreeing with him and answering back to what he said).
Anything else is not free speech.