Well Yordles, as you may or may not know, have surprisingly long lifespans. For instance, Poppy is said to have been around since the beginning of Demacia, a nation that is centuries old, and yet she still looks incredibly young. Taking this into account, although Kled is said to have been around for the beginning of Noxus, it is safe to assume that he is much older still, that is if Demacia and Noxus are roughly the same age. In one of the descriptions of the pair, Kled states that Skaarl is a drakalops, a creature that is immortal and indestructible, so even though it looks healthy and young it could easily be much, much older than Kled. In one of the discussion boards it was revealed that Kled and Skaarl had been together for so long that neither one could remember how they met. Now, considering the closeness of their relationship, their first meeting would probably have some significance, not something easily forgotten, so it would have happened a really long time ago. The first account of Kled in Noxian history, dating back towards the foundation of the empire, states that he was riding Skaarl at the time, which leads me to believe that they met earlier, probably even before the foundation of Noxus. Now, I do not know anything for certain, since I don't work for Riot, but I see maybe 3-4 possibilities for their first meeting.
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Kled was either lost or abandoned when he was young and had to try and survive on his own. He almost died and Skaarl found him and took care of him until he was well again. The two formed a bond and were inseparable from that time on. This would explain their friendship as well as Kled's distance from other Yordles and his general dislike and distrust of everyone.
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Skaarl may have shown up one day while Kled was out and about. Kled, being an ornery fellow, probably tried to shew Skaarl away but the creature kept following him, kind of like a friendly stray dog. Kled would've undoubtably tried to kill the creature but, of course, Skaarl is indestructible and immortal, and his attack would only serve to scare the creature. Skaarl would keep coming back and eventually Kled would give up and over time a bond would grow between the two. This makes sense because Kled is a distrusting and angry individual and so their friendship may have taken time to develop.
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Kled may have rescued Skaarl from humans trying to capture the poor creature, as a drakalops would undoubtedly be extremely valuable. Kled would be out and about, doing patrol, or whatever Kled was doing at this point in history, when he heard Skaarl's shrieks. Kled would come to the sound and have a bloody battle with the no good trespassers. Having saved Skaarl, Skaarl would grow fond of Kled, her rescuer, and would follow him around. eventually the the two would form an inseparable bond. This one is kind of a spin off the other two, except that Kled saves Skaarl in this one. This traumatic experience with humans would also explain Skaarl's cowardly, timid nature.
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Again, it is possible, that perhaps Kled stumbled upon Skaarl's egg. He undoubtedly thought of this as his next meal but when he goes to eat it it hatches and baby skaarl pops out. Kled, taken aback but still hungry, still tries to take a bite out of the tiny creature. Even as a baby, Skaarl's skin is too hard for Kled to chew. He throws the creature away and looks for other food, maybe mushrooms or berries. Kled finds a nice patch of berries and starts eating them, possibly resting under a tree, when Skaarl comes back. Even though Kled had tried to eat Skaarl, Skaarl has just been hatched and, much like birds, has imprinted on Kled, considering him her parent. Kled tries to ignore Skaarl but the little thing gives him the hungry puppy eyes and he ends up tossing Skaarl some berries. Kled grows fond of the little critter and thus their companionship is born. This is probably my favorite theory.
Like I said, I don't know anything for sure, but out of all the different theories, I feel that these 4 are among the most likely. It's not official but I hope this helps.