The primary source referenced for this claim is Relación de Inca Atahualpa y de don Francisco Pizarro, a manuscript written somewhere around 1535 by Juan de Betanzos, a Spanish explorer who accompanied Pizarro on his conquest of the Incas. De Betanzos conducted many interviews of surviving Incas, and ended up marrying Atahualpa's widow. This story has been brought to modern attention by Ricardo Palma (1838-1919), the former head of the Biblioteca Nacional in Peru.
The following is Palma's version, as recounted in pages 247-248 of his book Peruvian Traditions (this work is in the public domain):
Tradition has it that the captains Hernando de Soto, Juan de Rada, Francisco de Chavez, Blas de Atienza, and the treasurer Riquelme gathered every afternoon in Cajamarca in the room set aside as a prison for the Inca Atahualpa from November 15, 1532, when the monarch was captured, until two days before his unjustifiable execution, which took place on August 29, 1533.
In the room, the five men named and three or four others not mentioned in succinct and curious notes (that we had at hand, set down in an old manuscript preserved in the former Biblioteca Nacional), had the use of two crudely painted chess boards, set on a wooden chess table. The pieces were made of the same clay that the natives used for fashioning little idols and other objects of native pottery, which nowadays are dug up in huacas. Until the first years of the Republic the only pieces known in Peru were those made of ivory, sent by Philippine tradesmen for sale here.
The mind of the Inca must have been deeply preoccupied during the first two or three months of his captivity, for even though he seated himself every afternoon alongside Hernando de Soto, his friend and protector, he gave no sign of having realized how the pieces moved or how fortunes changed during a game. But one afternoon, during the end game of a match between Soto and Riquelme, Hernando made a move toward putting the knight into play, and the Inca, touching him lightly on the arm, said to him in a low voice: “No, captain, no . . . the rook!” Everyone was surprised. After a few brief seconds of reflection, Hernando played the castle, as Atahualpa had advised him, and a few moves later Riquelme experienced the inevitable checkmate.
After that afternoon, and always giving him the white pieces to play as a sign of respect and courtesy, Captain Hernando de Soto invited the Inca to play just one match with him, and after a couple of months the disciple was a credit to his teacher. They played as equals. The notes that I have mentioned tell how the other Spanish chess players, with the exception of Riquelme, also invited the Inca to play, but he always excused himself for not accepting, telling each of them through the interpreter Felipillo: “I play very little, and Your Grace plays a great deal.”
Popular tradition assures us that the Inca would not have been condemned to death had he remained untutored in chess. The people say that Atahualpa paid with his life for the checkmate that Riquelme suffered because of his advice on that memorable afternoon. In the famous council of 24 judges called together by Pizarro, Atahualpa was sentenced to the death penalty by 13 votes for and 11 against. Riquelme was one of the 13 who signed the death sentence.