The so-called 'Gestapo' is the nickname of the DSPLE (Division des Statistiques et de Protection de la Légion étrangère), based upon the name of the Nazi secret police during WW2. It's a colloquial term (so always put quotes around the name if your refer to DSPLE) due to their reputation of knowing everything about the recruits (or being able to).
Interview(s) with the DSPLE is basically a combination of a background check (where are you from, what did you do before trying to join the Legion, have you had any problems with the justice in your country, etc.) and a job interview (what are your reasons/motivations for willing to join the Legion, what are your qualifications/skills, etc.). Of course, there's some work done 'behind the scene' to verify what the candidate said. You may be summoned to additional interviews, if they want to cross-check, dig deeper into your declarations.
As Nick said, the results of this or these interview(s) are one of the elements taken into account by the final Selection Committee (in French Commission de Sélection) that meets once a week to review the results of the candidates, at least those who have made it up to this final stage (i.e. those who weren't sent back home or requested to leave earlier during the selection process). The members of the Committee/Commission decide who will become rouge (i.e. accepted in the Legion and given a 5-year contract before being shipped to Castelnaudary for basic training) and those who will be sent home (generally with an inapte définitif discharge).