The area currently known as Kaliningrad was formerly part of German East Prussia. It had been held by Germany since its foundation in 1870, and before that by Prussia. The area, known as Ostprussen (East Prussia) in Germany, has been held by Germans since the era of the Baltic Crusades.

During WW2, when the USSR was essentially rearranging the borders of Eastern Europe to its satisfaction, part of Ostprussen was given to Poland, and the remainder was annexed by the USSR. It was given to the RSFSR chiefly because there was no local ethnic minority who could credibly be given their own SSR. It was named Kaliningrad after Soviet head of state Mikhail Kalinin and given ASSR status (e.g. a semi-autonomous area of the RSFSR, like Chechenya or Kalmykia).

Since then Kaliningrad has become mostly ethnically Russian as Russians emigrated from other parts of the USSR, chiefly because of the strong local economy which was focused on shipbuilding and heavy industry. Kaliningrad is still, per capita, one of the wealthiest parts of Russia.

When the USSR fell apart there was some murmuring of reassigning Kaliningrad to Lithuania or Poland, but the fact that it was mostly ethnically Russian and still a major base for the Baltic fleet meant that Russia wasn't really prepared to countenance that. It seems that Kaliningrad's status isn't going to change, since there is neither popular or elite appetite to give it up.

There are some special visa arrangements for residents of Kaliningrad to visit Poland without needing to go through the usual visa arrangements but I am not aware of any special deal between Russia and Lithuania (Kaliningrad doesn't border Belarus or Latvia) easing the transport of goods between Russia proper and Kaliningrad. Having said that, Lithuania and Russia retain reasonably friendly relations and a close economic relationship, so I imagine there are not too many bureaucratic or economic hurdles for the movement of between Kaliningrad and the rest of Russia.