I mean, I get that they were trying to escape Communism and all that, but why were they trying to get into Berlin of all places? It was completely closed off, it doesn't seem like they would be able to go anywhere else except one city.
The position of the West Germany (FRG) was that all Germans could claim FRG citizenship and that Bonn was the temporary, legitimate government of Germany until a final peace treaty was signed between the German government and the Allies. This meant that any GDR citizen who made it to FRG territory could apply for citizenship in the FRG.
This left West Berlin as the easiest FRG territory for a refugee to access. GDR rail-lines and other transport networks did run through the divided city. Some refugees could buy tickets and flee during the trains' stopover in Berlin. Additionally, the large expansive nature of Berlin's urban geography made it easy to walk into West Berlin. There were a good deal of so-called "Sachsenganger", migrant GDR workers whose jobs were in West Berlin but were residents of the GDR, prior to 1961. Most refugees that reached West Berlin applied for asylum citizenship and were flown out of the city via the established air corridors. The erection of the Wall ended this migrant labor and also the last "easy' egress for an aspiring refugee. The Wall violently and at great costs to East German prestige did resolve the refugee crisis.
Nonetheless, crossing the Wall still was the best option for fleeing the GDR compared to the inner German border. The trickle of post-61 refugees would try both avenues, but Berlin remained disproportionately popular area for escape. The Wall was certainly a deadly strip of territory, but it was still nowhere near as militarized as the inner German border. Even though it was officially an "anti-fascist defensive rampart," the Wall was a tad less daunting than the other border available for refugees. The Western Berlin brigades were not much of a threat to the GDR and the Wall was less militarized than the inner border. The latter's fortifications were not only meant to stop refugees, but also to prevent a NATO drive into the GDR.