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The exact mechanics of how gathering works are a bit of a grey area, but I'll do my best to explain how the system works. The gathering system works off of a random number generator; first it picks the rarity of a stack of items you gather (the rarity of the item is displayed as a star in the item's tooltip, with rarer items having more stars), then it picks what specific item of that rarity is in the stack, and then how many of that item you will receive. In this way, items of a certain rarity will only compete with each other, and not with items of a higher rarity.

Suppose for example that you are Hunting. First the RNG will pick the rarity of your item, say one star. Then it will select one out of all available one-star items, like a bat. Then it will determine how many of said bats you will receive. Each stack of items you receive by gathering is determined by this process, so for each stack you have a chance of receiving a different item rarity, and a different item within that rarity.

Where does gathering level factor in, you might ask? Gathering level increases the pool of items that you have for each rarity, with better quality items becoming available at each level. To continue with the example above, you hunted and received bats as a one-star item. At higher levels, items like rabbits and quails (also one-star items) become available and begin to compete with bats for that slot. While there seems to be a bit of upgrading within the first few levels, generally speaking a new item becomes available at each gathering level starting at level 10.

As far as I understand, the chance of acquiring rarer items does not increase with your gathering level, just the quality of the items you can get at each rarity. Rare drops such as familiars and eggs are rare no matter what your level is.