"Pokémon Picross" is not so much "free-to-play" as it is "free-to-start."  The game is free initially and can remain free if you choose to keep it that way, but it can take a long time to progress if you do not invest any real money into it. 
As for what you're actually paying for, that would be "Picrites," which is the in-game currency.  Picrites, which are magical stones, are needed to open more puzzles and aspects of the game, as well as heal Pokémon after solving puzzles (recovery time can take hours to over a day per Pokémon). You cannot advance in the game without them. The main issue with playing for free the whole time is that you do not receive that many Pictrites as "drops" after accomplishing certain things in the game. To advance to other puzzles and areas and to have enough Pokémon in your party that can solve puzzles, you need hundreds to thousands of these. It could take days to weeks to earn enough for a single area without investing real money.
It is estimated that spending about $30 of real money will unlock everything in the game, which is about five times the price of an average Picross e-game. You should consider whether you plan to play this game short- or long-term when deciding on whether or not to invest real money into it.
* Image courtesy of Pokémon.com (http://www.pokemon.com/us/pokemon-video-games/pokemon-picross/)