The likelyhood is that Braun didn't develop something on their own. They likely used an off-the-shelf RF solution that operates in one of the license free ISM bands. So, it's likely operating around 430MHz or 915MHz or 2.4GHz. A spectrum analyzer would be helpful to ID that. The transmissions are likely to be short bursts, catching them on the analyzer will require careful setup unless you have some realtime capability in the analyzer. Modulation type is likely to be simple, such as OOK/ASK or FSK. If it is OOK/ASK, and you've identified the frequency, then the modulation pattern can be observed using zero span on the spectrum analyzer. Do really capture and analyze it, you'd ultimately want a signal analyzer (also known as a vector signal analyzer, not to be confused with a vector network analyzer). The VSA can capture/record the RF signal over time, then give you the ability to observe spectral, amplitude, frequency and phase changes vs. time.
Without this fancy equipment, you'll have to get creative. Building a simple broadband RF detector (using a short antenna and a few caps and signal diodes) and coupling it to a scope should at least give you an idea of the RF transmission duration, and possibly a view into the modulation (if it is a form of AM). That won't tell you about the frequency used though.
Is this information/feedback being provided "live", as you're brushing, or is it available only when the brush is put back on the stand. If the latter, the data transmission might not be RF at all...