Why did Russia give away Crimea in the first place?

Good question.  Discussed at length at the website of the Wilson Center:

Crimea was part of Russia from 1783, when the Tsarist Empire annexed it a decade after defeating Ottoman forces in the Battle of Kozludzha, until 1954, when the Soviet government transferred Crimea from the Russian Soviet Federation of Socialist Republics (RSFSR) to the Ukrainian Soviet Socialist Republic (UkrSSR)...

Of particular importance were the role of Nikita Khrushchev, the recent traumas inflicted on Ukraine, and the ongoing power struggle in the USSR... Khrushchev saw the transfer as a way of fortifying and perpetuating Soviet control over Ukraine now that the civil war had finally been won. Some 860,000 ethnic Russians would be joining the already large Russian minority in Ukraine...

The transfer of Crimea to the UkrSSR also was politically useful for Khrushchev as he sought to firm up the support he needed in his ongoing power struggle with Soviet Prime Minister Georgii Malenkov, who had initially emerged as the preeminent leader in the USSR in 1953 after Joseph Stalin’s death...

The earlier published documents, and materials that have emerged more recently, make clear that the transfer of Crimea from the RSFSR to the UkrSSR was carried out in accordance with the 1936 Soviet constitution... the main point to stress here is that it is incorrect to say (as some Russian commentators and government officials recently have) that Crimea was transferred unconstitutionally or illegally. The legal system in the Soviet Union was mostly a fiction, but the transfer did occur in accordance with the rules in effect at the time.

My excerpts probably don't do justice to the complexity of the arguments.  Those wishing to orate upon this at cocktail parties would do well to consult the source.