How did Kaliningrad Oblast come to be part of Russia?

It was part of East Prussia as the city of Koenigsberg: when Germany ceded East Prussia (and all territory east of the Oder-Neisse Line) at the end of World War 2, the Soviet Union proposed that they keep hold of Koenigsberg and its surroundings pending a final peace settlement. Article VI of the Potsdam Declaration provides for this.

The USSR and Poland then agreed a treaty in 1957 which demarcated the boundaries of the Kaliningrad Oblast as part of the Russian SFSR.

East Prussia had been evacuated in early 1945 and so the 'ethnic question' had been settled. Beevor (quoted in the article) states that the population of East Prussia dropped from 2.2 million in 1940 to 193,000 in 1945.

The Soviet Union presumably wanted a warm-water port in the Baltic Sea and so negotiated with Poland and the Allies to retain control of Koenigsberg after the war as there were no tricky ethnic questions to answer. The answer in short is that it was a spoil of war that was useful to the Soviets.

As Kaliningrad was part of the Russian Republic of the USSR, its territory became part of the Russian Federation upon breakup in 1991 and that has led to the modern situation.