The obvious answer is that they weren't. Or rather, the kind of exercise they did to train for war was not the same as the exercise needed to win competitions at the Olympics and other panhellenic games. Professional Greek athletes, whose entire lifestyle (including sleeping patterns, training and diet) was built around excellence in particular sports, would easily have been able to beat the average Spartan citizen at those sports, because Spartan exercise was aimed at general fitness and stamina, not specific skill. I explained this in more detail in this older thread which also has a link to this answer about the Spartan body.

The only Spartans who would be able to beat the athletes of other states would be those who spent their leisure time training for particular contests in addition to their state-mandated exercise. This doesn't seem to have been very common, and Spartan winners of combat sports at the Olympics are rare. (Although we do know of one Spartan general who was killed when his enemies caught him at discus practice.)

But Spartan citizens were all, by definition, leisure-class landowners, and so they often participated in (and regularly won) the chariot race, which was the quintessential sport for rich people. The main requirement was having the money to raise a team of 4 racehorses. Many Spartan citizens were very keen to show off to the rest of the Greeks that they could meet this requirement. In the 4th century BC, the Spartan princess Kyniska became the first woman to win the chariot race at the Olympics (despite not being allowed to attend).