Per the readme, "In the Gather Artifact mode, when you examine a place to explore and a stone is present, is displayed also the number required to level up that avatar."
I assume this means that when you see "9 more to level" associated with an avatar, winning that battle will make it "8 more to level" for the next battle in which that avatar is a reward, etc. No mention is made in the readme about the improvements to avatars as they level, however. Are these associated with special moves, or speed, or attack value, or defense? Will a special move last longer, or have a greater effect? Do improvements vary from unit to unit?