I think observing a single individual in both untreated and treated time intervals provides some amount of control. It's not a terribly strong control, of course; one can certainly imagine scenarios such as that they were recovering the whole time, but that recovery hadn't reached a point where the effects became noticeable until the point where they happened to start the tDCS. So in the strong sense of "know", you're correct: they can't know for sure.
But to say that it provides no information at all is also wrong. The posterior probability that the treatment was helpful is greater, given these observations both before and after treatment was started, than if, for example, treatment had been started immediately. How much greater is hard to quantify, but a very substantial observed improvement suggests a significant effect.
Everybody gets the control arm, followed by the treatment, but the time of the switchover is randomized, often in steps (hence the name). If people consistently recover at the same time relative to the treatment, but at different absolute times, then you can conclude the treatment works well.
I have no idea about this particular treatment, but I know that the only way to get an idea is rigorous double blind testing.
>ask brainstim01 how they know
That's an unanswerable question. He can't know. There is no way to know if an individual treatment worked for you in the absence of a control group. (Ignoring placebos which, sure, can and do "work")