Leveling up a currency-generating object increases the amount of passive income it generates. If you stand on such an object, the upper part of the screen has some yellow "Next upgrade" text that tells you what the new passive income will be if you raise the object to the next level. At certain levels (e.g., a Pine Tree's Level 26) there are very big increases in passive income. These levels with the big increases usually require a special items for leveling up (e.g., a Rain Cloud), not just berries or honey.
Likewise, leveling up a bear increases the percentage boost it provides to currency generation. If you stand on a bear, there is yellow text here also telling you what the new percentage boost will be if you level it up.
Finally, the village itself has a level. You can see the village level if you stand on an empty square. Your village level gives you a score boost while playing the word game. Also, some things (e.g., unlocking a new screen for your village) require a certain village level before you can do so.