Whether we in the present accept Jesus to be a real figure in history cannot retroactively influence whether he wrote things, so perhaps you mean "How is Jesus accepted as a figure in history without having left writings?"

And the answer is, it's completely unnecessary, given his culture, for him to have left writings. In fact, it would more interesting if he had than the fact that he did not. At the time, most stories and religious knowledge were shared orally, exclusively.

As an example of the difference, it is generally thought that the Christian gospels were not written down until near the end of the first century, and these were something you would expect to be important enough to write down.

For an overview of this information, check out The Bible: A Very Short Introduction by John Riches. Boston College also has some online discussion here.

There are references to Jesus' existence outside of the gospels as well, the most significant of which is written by Tacitus in Annals. An excerpt:

Consequently, to get rid of the report, Nero fastened the guilt and inflicted the most exquisite tortures on a class hated for their abominations, called Christians by the populace. Christus, from whom the name had its origin, suffered the extreme penalty during the reign of Tiberius at the hands of one of our procurators, Pontius Pilatus, and a most mischievous superstition, thus checked for the moment, again broke out not only in Judæa, the first source of the evil, but even in Rome, where all things hideous and shameful from every part of the world find their centre and become popular. Accordingly, an arrest was first made of all who pleaded guilty; then, upon their information, an immense multitude was convicted, not so much of the crime of firing the city, as of hatred against mankind.

I know linking to Wikipedia is generally discouraged but most of the sources I would provide are print, and there's a wealth of information on it in the article, including paths to more sources. Here is the article on Tacitus' recording of Jesus.

If you read any Ehrman, you'll realize just how differently literacy was regarded during that time period. Given what is stated of their origins, we would not expect either Jesus or any of his followers (Paul excepted) to be able to read or write. It was a very rare school of study available to only a handful. This, among other more important reasons, is part of how we know the Gospels were not actually written by Mark, Luke, etc. All the best research has shown that whatever we know of these people (that is to say, the gospels) comes from scholars writing at a much later date, and in a different language than the events unfolded in.