Amos, the same way as for studying anything else: you write down how you did it in enough detail that others can repeat your experiment. Remember, replicability is in the methodology at first - and IF there is replication of the results, you have evidence.

So, there are people doing reaction-time work: put people in front of computer screens and time their response to a stimulus. Change the stimuli, watch reaction-time change; you then have evidence for how thinking works if the reaction-time changes are predicted from the stimulus changes.
There are people doing fMRI work: put people in MRI machines with contrast (radioactive) sugars in them, give them things to think about, watch what lights up. If the same place lights up for others given the same tasks, you have evidence for where thinking happens.

And so on.

It really isn't different from studying anything else that appears unquantifyable to the general public... it just takes focus.