How old is this border?

Not so old! 1945!

The Oder-Neisse line is the arbitrary border drawn up and agreed at the Potsdam Conference (17 July to 2 August 1945). The physical geography suggesting this border involves two rivers, the Oder and Lusatian Neisse rivers. 

The Oder–Neisse line (Polish: granica na Odrze i Nysie Łużyckiej, German: Oder-Neiße-Grenze) is the international border between Germany and Poland. It was drawn at the Potsdam Conference in the aftermath of the Second World War and is primarily delineated along the Oder and Lusatian Neisse rivers in Central Europe, meeting the Baltic Sea to the north, just west of the Polish seaports of Szczecin and Świnoujście (German: Stettin and Swinemünde).

All prewar German territory east of the line and within the 1937 German boundaries (23.8% of the former Weimar Republic) were placed under International Law Administrative status, with most of it being made part of newly-Communist Poland. The small remainder, consisting of the territory surrounding the German city of Königsberg (now renamed Kaliningrad, in honour of Soviet head of state Mikhail Kalinin) in northern East Prussia, was allocated to the Soviet Union (as Kaliningrad Oblast of the Russian SFSR, today the Russian Federation) after the war (pending the final World War II peace treaty for Germany). The vast majority of the native German population in these territories fled, or were killed or expelled by force.

The Oder–Neisse line marked the border between the German Democratic Republic (East Germany) and Poland from 1950 to 1990. East Germany confirmed the border with Poland in 1950, while West Germany, after a period of refusal, finally accepted the border (with reservations) in 1970.

The bridges were burned
Now it's your turn, to cry

Cry me a river

The aftermath of World War in Europe included elements that are a strange mix of both punishment and re-construction of Germany. For the Soviet Union this was a sweet revenge. The new border meant a huge part of German territory became Poland. At the same time the eastern part of Poland, the Polish Ukraine, had been taken by the USSR. 

Borderlands
Incidentally, one of the possible origins for the word Ukraine is that it comes from slavic root words that translate as "borderlands".

The territorial changes of Poland

The territorial changes of Poland immediately after World War II were very extensive. In 1945, after the defeat of Nazi Germany, Poland's borders were redrawn in accordance with the decisions made first by the Allies at the Tehran Conference of 1943 where the Soviet Union demanded the recognition of the military outcome of the top secret Nazi-Soviet Pact of 1939 of which the West was unaware.

The same Soviet stance was repeated by Josef Stalin again at the Yalta Conference with Roosevelt and Churchill in February 1945, but a lot more forcefully in the face of the looming German defeat. The new borders were ratified at the Potsdam Conference of August 1945 exactly as proposed by Stalin who already controlled the whole of East-central Europe. 

Borders moving west . . .

The prewar eastern Polish territories of Kresy, which the Red Army had overrun during the Nazi-Soviet invasion of Poland in 1939 (excluding the Białystok region) were permanently annexed by the USSR, and most of their Polish inhabitants expelled. As a result of the Potsdam agreement to which Poland's government-in-exile was not invited, Poland lost 179,000 km2 (45%) of prewar territories in the east, including over 12 million citizens of whom 4,3 million were ethnically the speakers of Polish. Today, these territories are part of sovereign Belarus, Ukraine, and Lithuania.

In turn, postwar Poland was assigned considerably smaller territories to the west including the prewar Free City of Danzig and the former territory of Nazi Germany east of the Oder-Neisse line, consisting of the southern portion of East Prussia and most of Pomerania, Neumark (East Brandenburg), and German Silesia. The German population fled or was forcibly expelled before these Recovered Territories (official term) were repopulated with Poles expelled from the eastern regions and those from central Poland. The small area of Zaolzie, which had been annexed by Poland in late 1938, was returned to Czechoslovakia on Stalin's orders.