Other classes have easy explanations for how they came to be: fighters and monks trained, wizards studied, clerics and paladins prayed, sorcerers and oracles were born that way, and so on, so forth. Druids, however, do not seem to have any explanation for how, exactly, they became druids.
The pathfinder druid page gives the following class description (emphasis mine):
Within the purity of the elements and the order of the wilds lingers a power beyond the marvels of civilization. Furtive yet undeniable, these primal magics are guarded over by servants of philosophical balance known as druids. Allies to beasts and manipulators of nature, these often misunderstood protectors of the wild strive to shield their lands from all who would threaten them and prove the might of the wilds to those who lock themselves behind city walls. Rewarded for their devotion with incredible powers, druids gain unparalleled shape-shifting abilities, the companionship of mighty beasts, and the power to call upon nature’s wrath. The mightiest temper powers akin to storms, earthquakes, and volcanoes with primeval wisdom long abandoned and forgotten by civilization.
This seems to imply they are granted powers in a manner similar to clerics, but then the same page goes on to say:
Druids worship personifications of elemental forces, natural powers, or nature itself. Typically this means devotion to a nature deity, though druids are just as likely to revere vague spirits, animalistic demigods, or even specific awe-inspiring natural wonders.
Since druids who worship the concept of nature itself (or some aspect thereof) are just as prevalent as druids who worship any specific entity who may be acting as their patron, it seems more as if they are drawing power from nature, rather than being granted power. The overall druid fluff seems pretty clear on the idea that the amount of druidic power you have access to is based on your affinity with nature, (represented by character level) which explains why ceasing to revere nature causes you to lose that power.
The 9th level spell World Wave supports the "oneness with nature" concept, noting that:
creatures with even one druid class level (regardless of their type), are considered a part of the natural world
But the question remains of how one gains that first druid level in the first place. Do you just live as a hermit until you become one with nature and the powers develop on their own? Do you seek out and learn the druidic arts from an existing druid? On any race page, you can see druid is listed under the "Trained" age range, alongside alchemists, clerics, inquisitors, magi, monks, and wizards, which implies it is a time-consuming process to become a druid.
Mechanics-wise, it is perfectly feasible for, say, a tree-hugging monk to multiclass to druid, so there doesn't seem to be much of a barrier for entry into the class, beyond the characteristic devotion to nature, unless this is a handwave for player convenience.
Is there any official Paizo content that describes how a non-druid becomes a druid?
In the absence of Paizo content, I would also accept an answer based on D&D 3.x content, so long as it was still consistent with pathfinder lore.