Real Madrid published an official statement about FIFA's ban on their website, making their case and explaining why the club thinks they don't deserve said ban. Real also announced that they will appeal this ban in the upcoming days. Here's a fragment of that statement, you can read it all in Spanish here.
Real Madrid C. F. has today received the decision from the FIFA disciplinary committee, dated 23th July 2015, sanctioning the club with a transfer ban that prevents them from registering any players for the next two complete and consecutive registration periods as well as a fining the club 360,000 Swiss francs.
According to the decision, Real Madrid CF has been sanctioned for:
1.- Playing foreign players, under 18 years of age without being appropriately registered with the Real Federación Española de Fútbol (Spanish Football Federation): Absolutely untrue. All the players that play or have played in the youth teams in our club are and have been registered with the RFEF with prior notice before entering the field of play, as certified by the Spanish football federation to FIFA by means of a correspondence dated 17th April 2015, as the files show.
2.- Incorporating foreign players under the age of 18 years old, either for their first registration in Spain or through transfers from other foreign clubs, without fulfilling the regulatory requirements: Equally untrue. Real Madrid C. F. has always complied with FIFA regulation:
-With regards to players older than 12 years of age and younger than 18, has always complied with FIFA regulation requesting that the RFEF process (as they did) the corresponding authorisation from the FIFA Players' Status Committee.
-With regards to the players under 12 years of age, the RFEF (the only FIFA representative in Spain) established and communicated to Real Madrid C. F. (email dated 10th March 2014) that in relation to these players, they don't need a separate process to that of a Spanish under age player, which was ratified by FIFA to the RFEF by fax on 17th Aril 2014.
3.- Not communicating to the RFEF the players that play in the youth teams as FIFA considers that these teams are the equivalent to an academy: Also untrue. Although it is considered that the combined Real Madrid C. F. youth teams consist of an academy, despite not being independent legal entities, Real Madrid C. F. has always notified the RFEF the players who are registered in these teams, as can be proven by the fact that each and every one of them have the corresponding federative licenses, as stated in the document certified by the federation dated 8th April 2015.
As a result, Real Madrid C. F. will appeal this decision of FIFA in all the sporting incidents, considering them absolutely inappropriate.
Unless the appeal goes through successfully, Real Madrid will not be allowed to sign new players until the summer of 2017. Los Blancos can sign anyone they want this winter transfer market though, so they could get ready for that ban if they see fit.
Players on loan like Marco Asensio, Jesús Vallejo, Lucas Silva or Fàbio Coentrao will be able to make their returns to the team next summer and still contribute in the 2016-2017 season, as this ban only affects new registrations.