The Ancient Spartan society was based around the laws of Lycurgus, the rhetrae1, that were passed down through oral tradition. A possible explanation of why the laws were not written, and why Spartans didn't keep records in general, comes from Plutarch's The Life of Lycurgus:
[Plut. Lyc. 13.1] None of his laws were put into writing by Lycurgus, indeed, one of the so-called ‘rhetras’ forbids it. For he thought that if the most important and binding principles which conduce to the prosperity and virtue of a city were implanted in the habits and training of its citizens, they would remain unchanged and secure, having a stronger bond than compulsion in the fixed purposes imparted to the young by education, which performs the office of a law-giver for every one of them.
[Plut. Lyc. 13.2] And as for minor matters, such as business contracts, and cases where the needs vary from time to time, it was better, as he thought, not to hamper them by written constraints or fixed usages, but to suffer them, as occasion demanded, to receive such modifications as educated men should determine. Indeed, he assigned the function of law-making wholly and entirely to education.
[Plut. Lyc. 13.3] One of his rhetrae accordingly, as I have said, prohibited the use of written laws.
The fact that Lycurgus produced no writings is re-iterated at the end of the text:
[Plut. Lyc. 31.2] His design for a civil polity was adopted by Plato, Diogenes, Zeno, and by all those who have won approval for their treatises on this subject, although they left behind them only writings and words. Lycurgus, on the other hand, produced not writings and words, but an actual polity which was beyond imitation, and because he gave, to those who maintain that the much talked of natural disposition to wisdom exists only in theory, an example of an entire city given to the love of wisdom, his fame rightly transcended that of all who ever founded polities among the Greeks.
It should be noted however that the text starts with pointing out that Lycurgus was an enigmatic figure, and that not much can be said definitively about his life:
[Plut. Lyc. 1.1] Concerning Lycurgus the lawgiver, in general, nothing can be said which is not disputed, since indeed there are different accounts of his birth, his travels, his death, and above all, of his work as lawmaker and statesman; and there is least agreement among historians as to the times in which the man lived. Some say that he flourished at the same time with Iphitus, and in concert with him established the Olympic truce. Among these is Aristotle the philosopher, and he alleges as proof the discus at Olympia on which an inscription preserves the name of Lycurgus.
Again according to Plutarch, Lycurgus laws were of divine origin, having been passed at him from the Oracle at Delphi:
[Plut. Lyc. 5] He was convinced that a partial change of the laws would be of no avail whatsoever, but that he must proceed as a physician would with a patient who was debilitated and full of all sorts of diseases; he must reduce and alter the existing temperament by means of drugs and purges, and introduce a new and different regimen. Full of this determination, he first made a journey to Delphi, and after sacrificing to the god and consulting the oracle, he returned with that famous response in which the Pythian priestess addressed him as ‘beloved of the gods, and rather god than man,’ and said that the god had granted his prayer for good laws, and promised him a constitution which should be the best in the world.
Apparently claiming divine origin for his laws wasn't enough, and Lycurgus asked the Spartans to swear an oath that they would abide by his laws until he returned from another journey at Delphi. When they did, he travelled to Delphi, and after he was re-assured about his laws, he killed himself, binding the Spartans to follow his laws ad infinitum:
[Plut. Lyc. 29.2] Accordingly, he assembled the whole people, and told them that the provisions already made were sufficiently adapted to promote the prosperity and virtue of the state, but that something of the greatest weight and importance remained, which he could not lay before them until he had consulted the god at Delphi. They must therefore abide by the established laws and make no change nor alteration in them until he came back from Delphi in person; then he would do whatsoever the god thought best.
[Plut. Lyc. 29.3] When they all agreed to this and bade him set out on his journey, he exacted an oath from the kings and the senators, and afterwards from the rest of the citizens, that they would abide by the established polity and observe it until Lycurgus should come back; then he set out for Delphi.
On reaching the oracle, he sacrificed to the god, and asked if the laws which he had established were good, and sufficient to promote a city's prosperity and virtue.
[Plut. Lyc. 29.4] Apollo answered that the laws which he had established were good, and that the city would continue to be held in highest honour while it kept to the polity of Lycurgus. This oracle Lycurgus wrote down, and sent it to Sparta. But for his own part, he sacrificed again to the god, took affectionate leave of his friends and of his son, and resolved never to release his fellow-citizens from their oath, but of his own accord to put an end to his life where he was. He had reached an age in which life was not yet a burden, and death no longer a terror; when he and his friends, moreover, appeared to be sufficiently prosperous and happy.
Herodotus, a much earlier source, gives a slightly different account, in the first book of his Histories:
[Hdt. 1.65.2] Before this they had been the worst-governed of nearly all the Hellenes and had had no dealings with strangers, but they changed to good government in this way: Lycurgus, a man of reputation among the Spartans, went to the oracle at Delphi. As soon as he entered the hall, the priestess said in hexameter:
[Hdt. 1.65.3] “You have come to my rich temple, Lycurgus, A man dear to Zeus and to all who have Olympian homes. I am in doubt whether to pronounce you man or god, But I think rather you are a god, Lycurgus.”
[Hdt. 1.65.4] Some say that the Pythia also declared to him the constitution that now exists at Sparta, but the Lacedaemonians themselves say that Lycurgus brought it from Crete when he was guardian of his nephew Leobetes, the Spartan king.
[Hdt. 1.65.5] Once he became guardian, he changed all the laws and took care that no one transgressed the new ones. Lycurgus afterwards established their affairs of war: the sworn divisions, the bands of thirty, the common meals; also the ephors and the council of elders.
[Hdt. 1.66.1] Thus they changed their bad laws to good ones, and when Lycurgus died they built him a temple and now worship him greatly. Since they had good land and many men, they immediately flourished and prospered.
Although Herodotus claims a Cretan origin2 for Lycurgus' laws, the Oracle and divine favour still feature prominently.
Lycurgus status remains enigmatic, although he is mentioned by name in several other sources, including Thucidides History of the Peloponnesian War, Plato's Laws, Phaedrus, and the Republic, and Aristotle's Rhetoric. Aristotle also alludes to the fact that the Spartans weren't big fans of learning in general:
[Aristot. Rh. 2.23.11] the Lacedaemonians, by no means a people fond of learning, elected Chilon one of their senators;
Even if Lycurgus never existed in the first place, it is clear that he enjoyed a legendary hero cult status in Ancient Sparta and his retrae were thought to be divinely inspired, and thus followed dogmatically. All contemporary and near contemporary sources suggest that the Spartan society relied almost exclusively on oral tradition. The only archaeological finds that go a bit against the theory that Ancient Spartans kept no written records are a few inscriptions found in the Evrotas valley, no literary works survive, assuming they ever existed. These limited finds, however, by no means compare to the wealth of literary evidence we have for the rest of the Ancient Greek world, it's possible that they were simply minor transgressions.
1 A common translation would be "sayings".
2 Crete's own legendary law-giver, Minos, was thought of as a son of Zeus, and the Cretan origin Herodotus suggests may imply that the retrae were based on the divine laws of Minos. Plutarch also mentions that Lycurgus visited Crete, prior to presenting the rhetrae to the Spartans.