These terms came to the crypto market from the traditional stock market trading. In both markets the words "long" and "short" convey whether a trade originates from buying first or selling first. A long trade means that you buy an asset with the intention to sell it later, when the rate goes up, and make a profit. A short trade means that you sell an asset, before buying, with the intention to re-buy it later, when the rate goes down, and make a profit.
Day traders often will use the terms "buy" and "long" interchangeably. Similarly, some trading software has a trade entry button marked "buy," while others trade entry buttons marked "long."
Another pair of terms often used interchangeably with the "long" and "short" is the "bull" and "bear."