I apologize if this answer seems a bit pin-headed, but I see 3 deleted answers that "don't answer the question" because they do not describe the origin of the term. I am positive that by describing the origin of the term, this answer will be regarded as unsuitable for this gaming q&a site... but here goes anyway.
Thirsting is a present participle form of thirst (verb) and it means: feeling a need to drink. There are other uses beyond the common one which generally mean: to desire. Thirsting for knowledge. Thirsting for blood - blood thirsty. Sometimes the term is used to indicate a desperate mindset, that thirst can override good decision making.
The earliest use of the term that I have found so far that means killing a downed opponent is from 2018-11-28 in a Fortnite Battle Royale subreddit.
None of those reason include getting thirsty as !#?$ just trying to get a kill before the enemy shooting you kills you. Absolutely hate the people who ignore trying to win the fight and just thirst for that 1 kill before they die.
I looked for other uses of the term in other games and only found various discussions of thirst systems throughout 2017-2018. Thirst systems usually require the character to drink something periodically to avoid negative effects.
Meaning in Fortnite BR
Thirsting has two meanings. In both, a downed opponent is focused while there are other standing opponents.
The first meaning is tactical. By focusing on a downed opponent, the player does not contribute to his teams success in the fight while also risking a trap. It is seen as a tactical mistake to prioritize kill count over winning the engagement.
The second meaning is strategic. If both groups kill downed opponents, whoever wins will be left with fewer teammates for the next fight. If both groups ignore downed opponents, whoever wins will be at full strength for the next fight and more likely to win the round. By following this short term cooperative policy, a longer term competitive advantage is gained.