Why did the Spartans learn to read and write if they weren't allowed to keep records?

Indeed, Plutarch records that the Spartan education in reading and writing was minimal:

The boys learned to read and write no more than was necessary. Otherwise their whole education smart obedience, perseverance under stress, and victory in battle.

^{(On Sparta Penguin Classic p. 21)}

Presumably, the ban on writing did not cover everything; indeed, earlier in the same book Plutarch describes it as specifically a ban on writing laws.

SHORT ANSWER

There are many documented examples of Spartans using writing, including the recording of laws and oracles, and the literacy rate amongst women may well have been higher than in any other ancient Greek city state. Several ancient sources (including Herodotus and Thucydides) contradict the assertion that the Spartans did not keep written records, and they had plenty of uses for writing, especially letters and treaties during the Peloponnesian War.

It is important to note that, aside from works by more recent academics, much of what has been written about Sparta has been exaggerated or sensationalized and, even in ancient times, the Spartans themselves did not discourage many of these 'myths'.

DETAILED ANSWER

In his book Spartan Law, Prof. D. M. MacDowell argues convincingly that one should not interpret the claim that Spartans did not write down their laws too strictly, especially when it comes to amendments to the original constitution. Plutarch quotes from a text of one of the laws. On this, MacDowell says:

...it seems clear...that it is a genuine early Spartan document. Nearly all modern scholars accept it as such,...

...he [Plutarch] goes on to quote a poem of [the Spartan] Tyrtaios which refers to the law with the rider. Thus it appears that at least one of 'the laws of Lykourgos' was written by the time of Tyrtaios, before the end of the seventh century.

Further evidence concerns the procedure for legislation proposed by King Agis IV (died 241 BC), 'who claimed to be a champion of tradition' (MacDowell):

From this evidence it appears that the procedure had three stages. First, the proposal was put in writing and brought forward by one or more of the ephors.

There are a number of references in Herodotus to the Spartans not only being literate (see example of Gorgo below) but also keeping records. For example, they kept records of oracles from Delphi.

One argument which has been used as evidence that the Spartans had little interest in literacy is that there was a lack of Spartan literary figures during the classical period. Compared to Athens, true enough, but Athens

was probably atypical in its abundance of writers....Despite Sparta’s lack of native writers, its citizens utilized literacy in a number of ways and on a variety of levels that together reveal the written word’s important place in their society.

Source: Ellen G. Millender, 'Spartan Literacy Revisited'. In 'Classical Antiquity Vol. 20, No. 1' (April 2001)

Millender goes on to point out that more recent scholarship has observed that

the epigraphical and literary evidence suggests that the Spartans made use of a number of public documents, many of which were stored in a rudimentary archival system in Sparta.

The Spartans

not only stored oracles and Tyrtaeus’ poetry but also kept records of athletic victories and copies of treaties.

Where were these archives? This is uncertain:

While we have no early evidence of a central building designated as the official records office, the ancient sources suggest that the houses of the kings served, perhaps unofficially, as city archives at least through the fourth century and probably longer.

Source: Millender

In addition to the kings' houses, at least some records were kept in temples. Thucydides notes that copies of treaties during the Peloponnesian War were publicly displayed. He also mentions many treaties (in the Spartan dialect, Doric) and

Spartan commanders in the History communicate by letter to one another and back to Sparta (8.33.3; 8.39.2; 8.45.1). Athenians, too, send letters (1.137.3–4; 7.8.2; 8.51.1), but the impression persists that in Thucydides, particularly in Book VIII, Spartan commanders communicate with their superiors at home more often than do their Athenian counterparts.

Source: Paola Debnar & Paul Cartledge, 'Sparta and the Spartans in Thucydides'. Chapter 22 in A. Rengakos & A. Tsakmakis, 'Brill's Companion to Thuydides' (2006).

Spartans were not always brief either, but one fine example of Spartan brevity (and helplessness when left without a leader), is a message which was intercepted by the Athenians after they had defeated the Spartan navy at the Battle of Cyzicus in 410 BC. The Spartan admiral, Mindaros, was killed during the battle. Xenophon records this in Hellenika thus:

A letter that had been sent to Lacedaemon by Hippokrates, Mindaros' vice-admiral, was captured and brought to Athens. It said, "Ships gone, Mindaros dead, men starving; at our wits' end what to do."

In another example of a letter, Paul Cartledge, in his book The Spartans, drawing on Herodotus, writes of a message sent by the exiled former King Demaratus which puzzled all but Gorgo, wife of King Leonidas I:

A messenger arrived in Sparta bearing an apparently blank wax tablet (two leaves of wood, covered with wax and folded together). 'No one,' Herodotus relates, 'was able to guess the secret' - no one except Gorgo, that is. She calmly told the authorities that if the wax were scraped off, they would find the message written in ink on the wood beneath, and so indeed it proved.

Interestingly, Spartan women were given a formal education and could at the very least read, unlike their counterparts in other Greek city states (including Athens) who received no formal education (though some girls from wealthy families were taught to read and write). Evidence of this can be found in Spartan sayings recorded by Plutarch:

When some woman heard that her son had been saved and had escaped from the enemy, she wrote to him: 'You've been tainted by a bad reputation. Either wipe this out now or cease to exist.'

A final note - when dealing with Sparta, beware the Spartan myth. Some of what one finds outside of modern academic work on Sparta can be very misleading. As Cartledge notes,

The Spartan myth was persuasively labelled a 'mirage' in the 1930s, by the French scholar Francois Ollier, because the relation between the myth and reality was and is sometimes so hard to perceive without distortion.

Other sources:

Anne Pearson & The British Museum, 'Ancient Greece'

The Landmark Xenophon's Hellenika (trans: John Marincola)