Nobody knows for certain where the scoring system in tennis came from. However, there is a certain logic behind it. It makes sense to count each game as one game, so if each point also counted as one point, then the score could be confusing, For example, if one side is winning 3-1, 1-2, then it could be difficult to keep track of the score. Counting points by 15 removes that confusion, but why choose 15? Why not some other number, like 10, or maybe even a letter?
There are other number systems that seem equally strange, but have a different logic. For example, there are 360 degrees in a circle, and there are 60 minutes in an hour. Those numbers were chosen since they can be divided by many other numbers (including 15) without having a fractional part left over. These days, it would make more sense to divide a circle into 100 degrees and it would make more sense to divide an hour into 100 minutes. Better still, divide one day into 10 or 100 hours and divide each hour into 10 or 100 minutes. But we are stuck with ancient traditions, nobody is complaining, and it is likely to stay that way for a long time.
Units used to measure mass, length and currency also have long histories. The craziest is the old English system where 12 inches = 1 foot, 3 feet = one yard, 12 pence = one shilling, 20 shillings = one pound, and so on. Given that tennis had its origins in France during the Middle Ages, it is likely that the scoring system was based on an old French counting system. In French, a quinzaine is a quantity of 15, in the same way that a dozen is a quantity of 12. The French word for 15 is quinze (nothing to do with quince fruit). Counting by 15 was therefore common, and it included an old coinage system where a gold crown coin was valued at 60 sous, although the system changed many times as the various kings of France kept changing the monetary system. Big bets were placed by the rich and famous on games of tennis in the Middle Ages, with 15 sous awarded to the winner of each point. At least, that is how some historians have interpreted the scoring system. It is a plausible explanation, but by no means certain.