My answer would be basically the same as Brian Z, but since I spent the last hours getting the original of what he quoted and preparing a at least readable translation (with the help of Google Translate), I will just dump it here.
Murray seems to have shortened even the one sentence he quoted substantially. If you read the text in its entirety, you can see how chess is treated as a general allegory of human life and sinfullness. In this perspective likening death and a checkmate is a quite obvious choice, even without anthropomorphing death.
The text is the second half half of a chapter called "An admonition against the vacant games of chance etc." (De admonitione vacantium ludis alearum etc.) of the book Summa collationum by John of Wales (in this 1516 print it starts at the bottom of the right column on the linked page). I quote the latin text from Antonius van der Linde, Quellenstudien zur Geschichte des Schachspiels, Springer 1881 pp. 65-67.
Mundus iste totus quoddam scaccarium est cujus unus punctus est albus et alius niger, propter duplicem statum, vitae et mortis, gratiae et culpae. Familia hujus scaccarii sunt homines hujus mundi qui omnes de uno sacculo matutino extrahuntur et collocantur in diversis locis hujus mundi et singuli habent diversa nomina. Unus dicitur rex, alter regina, tertius rochus, quartus miles, quintus alphinus, sextus pedinus; unde versus: Rex, Rochus, Alphinus, Miles, Regina, Pedinus. Istius autem joci conditio est talis ut unus alium capiat et cum ludum compleverint, sicut de uno loco et sacculo exierunt, sic in unum locum reponuntur, nec est differentia inter regem et peditem pauperem; quia similiter in unum dives et pauper; et Sap. 7 v. 5: unus est introitus ad vitam et similis exitus. Et saepe contigit quod, quando familia scaccarii reponitur in sacculum rex inferius collocatur et sic quando transeunt ab hac vita hujus mundi majores in inferno sepeliuntur et pauperes in sinum Abrahe deportantur, exemplo divitis et Lazari.
In isto etiam ludo rex vadit circumquaque directe et capit undique semper directe, in signum quod rex omnia juste capiat et in nollo omissa justitia, omnibus exhibenda, obliquare debet, sed modo quidquid principi placet juris habet vigorem. – Regina sive domina quae dicitur Ferze capit et vadit oblique, quia cum avarissimum sit genus mulierum, quidquid capit, nisi mere detur ex gratia, rapina est et injustitia. – Rochus est justitiarius perambulans totam terram, directa tamen linea, ita quod nihil oblique capiat muneribus corruptus, sed omnia juste corrigat nulli parcens, sed contra de illis jam verificatur illud Amos prophetae: Convertisti in amaritudinem judicium et fructum justitiae in absinthium. – Miles vero in capiendo dua puncta transit directa et tertium obliquat; in signum quod miles et domini terreni paterunt juste capere redditus debitos et justas emendas a delinquentibus secundum exigentiam delicti, sed tertium punctum obliquat cum talliagia et injustas actores extorquent a subditis. – Alphini vero cornuti sunt episcopi non ut Moyses ex colloquio divino sed potius regio imperio vel prece aut pretio sublimati. Isti alphini oblique currunt et capiunt tria puncta pertranseundo, quia fere omnes praelatos pervertunt odium et amor et mulierum favor, ne delinquentes reprehendant et contra vitia latrent, sed potius pro annuo censu peccata ad fermam tradunt ut sic dyabolum ditent, unde qui debuerunt vitiorum extirpatores esse, jam per cupiditatem facti sunt vitiorum promotores et Dyaboli procuratores. – Pedinus vero pauperculus est qui incedendo semper vadit directe in sua simplicitate, sed quando capere vult obliquat sic semper dum in sua simplicitate et paupertate consistit directe vadit, sed cum quaerit aliquid temporale vel honorem consequitur, statim mendatiis, perjuriis, adulationibus et favoribus obliquat. Quosque ad gradum superiorem scaccarii perveniat et tunc duo puncta pertransit, tertium obliquando. Sic pauper cum elevatur statim perverse incedit, quia asperius nihil est humili cum surgit in altum.
In isto scaccario Dyabolus dicit eschack, insultando aliquem et percutiendo peccati jaculo; qui sic percussus, nisi citius dicat linqueret ad penitentiam recurrendo, dicit Dyabolus ei Math, animam suum ad tartara deducendo a quo nec liberabitur prece vel pretio, quia in inferno nulla est redemtio. Et sicut venator diversos habet canes ad capiendas bestias diversas, sic Dyabolus et mundus diversa habent peccata quibus diversimode homines illaqueant, quia omne quod est in mundo aut est concupiscentia carnis aut oculorum aut superbia vitae.
This whole world is a kind of chess game, whose fields are alternately white and black, because of the twofold state of life and death, grace and guilt. The family of chess are the people of this world, who are all drawn from one bag in the beginning and placed in different places of this world and each one has different names. One is called a king, another a queen, a third a rock, a fourth a knight, a fifth an elephant, and a sixth a pawn; whence the line: King, Rock, Elephant, Knight, Queen, Pawn. Now the rule of this game is such that one takes another, and when the game is finished, just as they left one place and bag, so they are put back in one place, and there is no difference between a king and a poor pawn; because the rich and the poor become one and the same; as says Wisdom 7 v. 5: One is the entry into life for all, and in one same way they leave it. And it often happens that, when the chess family is placed in a bag, the king is placed at the bottom, and so when they pass from this life the masters of this world are buried in hell and the poor are carried into Abraham's bosom, after the example of the rich man and Lazarus.
In this game the king goes round and round directly, and seizes from every side always directly, as a sign that the king should seize everything justly, and if unwilling to bring justice to all, he must bow, but only to what the law prescribes to him. – The queen or lady called Ferze takes and goes sideways, because women being a most avaricious race, whatever they take, unless it is given purely out of grace, is robbery and injustice. – The rock is a righteous man walking through the whole earth, yet a straight line, so that he does not take anything obliquely, corrupt by gifts, but corrects everything justly, sparing no one, but the saying of the prophet Amos truely applies to him: You have turned judgment into bitterness and the fruit of justice into wormwood. – The knight, in capturing two fields, passes directly and passes the third obliquely; as a sign that the soldier and the lords of the land are permitted to take justly the due rents and just compensations from the offenders according to the demand of the crime, but the third field is oblique when the collectors and unjust actors extort from their subjects. – As for the elephants, these horned bishops came not, like Moses, from a divine proclamation, but rather by the command of the kingdom, or by prayer, or by the price of potent spirits. These elephants run obliquely and capture three fields in passing, because almost all the prelates pervert hatred and love and the favor of women, in that they do not rebuke offenders and bark against vices, but rather deliver sins to the farm instead of the annual census, so as to enrich the devil, whence those who ought to be exterminators of vices, have already become promoters of vices and agents of the Devil through greed. – The pawn is a poor man who always goes directly in his simplicity when he advances, but when he wants to capture he turns aside, so he always goes directly as long as he remains in his simplicity and poverty, but when he seeks something temporal or obtains honor, he immediately turns aside with lies, perjury, flattery and favors. Each of them reaches the upper level of the chess board and then passes through two fields, bypassing the third. Thus, when the poor man is exalted, he immediately walks in a perverse way, because nothing is harsher than the lowly when he rises to a high place.
In this chess game the Devil says eschack to man, and strikes him with the arrow of sin; he who is thus smitten, unless he immediately says that he should leave by resorting to penance, the Devil says to him Math, leading his soul to Tartarus, from which he will not be delivered either by prayer or money, because there is no redemption in hell. And as a hunter has different dogs to catch different beasts, so the devil and the world have different sins by which they entice men in different ways, because everything that is in the world is either the lust of the flesh or the eyes, or the pride of life.
