I assume you are talking about the events of mid to late 1989, which saw several thousand East Germans – not just Berliners – move to Prague in an attempt to secure passage to West Germany at their embassy there. In fact, it was not just Prague which had to deal with the issue, the FRG embassies in Warsaw and Budapest also experienced this sudden arrival of thousands of Germans. Indeed, the latter would have been the original destination for the Germans, as Hungary had started to remove its strict border controls and barriers with Austria in May 1989, opening the door for the East Germans to move to West Germany via the neighboring country to the south.
Back in Prague, there were three waves of East Germans leaving for West Germany between late August and early November, until the fall of the Berlin Wall and even a short time afterwards. According to Herman Huber, the former FRG ambassador in Prague, some East Germans have sought refuge at the embassy already in early 1989. But it was only in August when their numbers grew to significant amounts.
By August 19, 123 East Germans were present at the embassy, with dozens more arriving every day. At first, these were lodged in the embassy itself – the Lobkowicz Palace in Prague's Lesser Town, not to be confused with a palace of the same name located at the Prague Castle – a sizeable building on its own. When its capacity became insufficient, they were housed in the orangery, a sort of a fancy greenhouse, on the grounds of the palace complex.
However, the influx of people showed no signs of stopping. Despite closing its doors for the public on the 23rd on paper, the FRG embassy did not prevent people from finding their way to the embassy grounds, often by climbing over the walls and fences, as you can see in this news report. A tent city was quickly erected in the palace gardens, and even a makeshift school was set up for the children. Embassy workers soon saw their priorities shift from their usual work to taking care of the refugees with the help of the Red Cross and even some of the locals who brought food, hot beverages or blankets to the embassy.
The German embassy even has a page dedicated to these events (in Czech). A long time embassy worker Ivana Vlnasová recalls: "We only cared about the refugees. We brought them food, took care of their everyday needs and hygiene." As you can imagine, the situation was difficult at best. At one point, there were just twenty two toilets available, the capacities of the field kitchens limited, people slept in shifts, and the weather was getting worse.
For several weeks, there seemed to be no immediate solution on the horizon. It was only at the end of September that, as a humanitarian gesture, the East German leadership acquiesced to the free travel of East Germans at the West German embassy in Prague. On September 30, FRG foreign minister Hans-Dietrich Genscher broke the good news to the crowds at the embassy in what is now a somewhat iconic scene, and by October 1, the first wave of refugees, over six thousand people, embarked on special trains to West Germany.
Even more would soon follow. Thousands more, in fact, would arrive in the coming days, not only camping on the embassy grounds, but in the surrounding streets as well – you might be interested in a couple of photographs. Late on October 4, the second wave of over eight thousand East German refugees would depart from Prague. By now, on the 3rd, East Germany had again restricted travel to Czechoslovakia, but that would only prove to be a temporary setback as the leadership would give in to pressure and free travel resumed on the first of November.
Thus the third and final wave wave started, with 6500 East Germans departing on November 4. Then, late at night on November 9, the Berlin Wall finally fell and other East-West German border crossings were soon opened. With the way now both literally and figuratively open, some East Germans still kept moving to West Germany via Prague on the 10th and in the coming days, but by then, crisis was now over.
While serious, these events also had a rather funny aftermath. Desperate to move to West Germany, many East Germans were willing to abandon their possessions. This resulted in hundreds of Trabant cars, a bit of a legendary brand in the Eastern Bloc, being abandoned in the streets of Prague. Some got broken into, some got stolen, some got towed, and a large number of them were repossessed by East German authorities.
Since 2001, a copy of a 1990 sculpture by Czech artist David Černý commemorating these events stands in the palace garden: a four-legged Trabant dedicated to "the memory of many thousand Germans of the GDR who in the summer and fall of 1989, via the embassy in Prague, sought and found their way to freedom." Its title, "Quo vadis?" – "Where are you going?"