The answer to this is a little convoluted and the answer is just what is "believed" to have been the reason behind the scoring numbers, but since tennis is a game that dates back to the 19th century, it's tough to know for sure.
Anyway, here goes...
Think of a clock face that is divided up into quadrants - which would give you the 15, 30, 45, and 60 intervals. The problem with using that to score tennis games though, is that to win a game, you have to win by 2 points if the score reaches deuce (40-40 or "40 all"). So it is believed that the first 3 points would advance the hand to 15, 30 and 40 - and then the next 2 points would advance the hand by 10 minutes - first to 50, then to 60 - still keeping the game finished at the 60 mark. If the (serving) player failed to win 2 points in a row after deuce, the hand would move back to 40 and the score would be at deuce again.
See here for some additional theories behind the scoring nomenclature.
The best explanation I could find of this is from Wikipedia:
The origins of the 15, 30, and 40 scores are believed to be medieval French. It is possible that a clock face was used on court, with a quarter move of the hand to indicate a score of 15, 30, and 45. When the hand moved to 60, the game was over. However, in order to ensure that the game could not be won by a one-point difference in players' scores, the idea of "deuce" was introduced. To make the score stay within the "60" ticks on the clock face, the 45 was changed to 40.
Therefore, if both players have 40, the first player to score receives ten and that moves the clock to 50. If the player scores a second time before the opponent is able to score, they are awarded another ten and the clock moves to 60. The 60 signifies the end of the game. However, if a player fails to score twice in a row, then the clock would move back to 40 to establish another "deuce".
It makes very well sense that this is the case.
I've also heard the story that the very first tennis players (we shall call them Adam and Eve) used a clock's minute hand to keep score (15, 30, 45). However, the clock was slightly broken and the hands were a bit loose. So when Eve went to set the score to 45, the hand slipped and fell back down to 40! And that has been the way they score tennis ever since.
As for the history of tennis scoring, there are two background stories:
- That it has its origin in medieval numerology. The number 60 was considered to be a "good" or "complete" number back then, in about the same way you'd consider 100 to be a nice round figure today. The medieval version of tennis, therefore, was based on 60--the four points when 15, 30, 45 (which we abbreviate to 40) and 60, or game.
- The system may be based on the presence of a clock face at the end of the tennis court. A quarter move of the appropriate hand was made after each rest, with the score being called as 15, 30, or 45. As the hand was moved to 60, this was the game. This didn't explain a score of 40, however.
The theories of the clock are as I understand it correct in that the score was kept on a clock using the 15, 30 and 45 minute marks ... so why 40? To understand this it is good to realise that this all was established in France and when the scores were called out they would have called quinze (15), trente (30) and quarante-cinq (45) .. however quarante-cinq is a bit of a mouthful (3 syllables instead of one) compared to the other two so it was abbreviated to simply "quarante" or 40. When the rules became standardised in the late 19th century in both the US and Britain ..it became simply ..15, 30 , 40
The modern rules of lawn tennis were defined by a British Army, Major Wingfield.
He used the calibre of cannon used on board ships to define the scoring system - the standard cannon were 15 pounders, 30 pounders and 40 pounders. He retained 'love' (based upon 'oueff - 'egg' in French i.e. looking like nought) and deuce (based upon 'deux' in French, meaning two more points needed).
I know this because I am his great, great grandson.
Another theory on this scoring system (15, 30, and 40 scores) is:
The scoring nomenclature came from the French game jeu de paume (a precursor to tennis which initially used the hand instead of a racket). Jeu de paume was very popular before the French revolution, with more than 1,000 courts in Paris alone. The traditional court was 90 ft (pied du roi) in total with 45 ft on each side. When the server scored, he or she moved forward 15 ft. If the server scored again, he or she would move another 15 ft. If the server scored a third time, he or she could only move 10 ft closer.
Cited Source: Françoise Bonnefoy (1991). Jeu de Paume: History. Réunion des musées nationaux. p. 42. ISBN 9782908901016.
Since a third score would put the server right at the net, 10 feet was the last bump forward. Source