(I've been wanting to answer this question since it was posted, but I couldn't find the time to formulate a proper response until now. I hope that you don't mind the delay and that you find this answer useful.)

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What motivated ancient historians is relatively easy to answer, since they had a habit of explaining why they wrote their work in the first few pages. First though, I'd like to remark that being a historian was not a profession as such in ancient times. The historians we know of were either wealthy enough to spend their time writing (usually at the end of their career) or had a patron who was supporting them. They wrote their works out of interest or to flatter their patron. Usually the work was intended to lead to understanding about current events. Often it had a moral component to it, which sometimes even dominated the work. Xenophon's Cyropaedia for instance is ostensibly a biography of Cyrus the Great, but it is generally understood to give an idealized description, perhaps even to the point of describing an ideal ruler instead of the person Cyrus the Great.

Let's look at what some of the most well-known ancient historians say about their own work.

• Herodotus in his Histories wanted to record what were the grounds for the feud between the Greeks and the Persians.
• Thucydides, writing on the Peloponnesian War, said he started writing his history at the start of the war, because he believed "that it would be a great war, and more worthy of relation than any that had preceded it" (1.1.1).
• Xenophon's Anabasis is essentially a memoir of his time with the Greek mercenaries hired by Cyrus the Younger. His Cyropaedia (mentioned above) was written to understand how Cyrus the Great became to be such an excellent ruler: "Believing this man to be deserving of all admiration, we have therefore investigated who he was in his origin, what natural endowments he possessed, and what sort of education he had enjoyed, that he so greatly excelled in governing men. Accordingly, what we have found out or think we know concerning him we shall now endeavour to present." (1.1.6)
• Polybius - a Greek who went to Rome as a hostage and became a friend of Scipio Aemilianus - wrote his Histories to understand how Rome had succeeded in maintaining an empire whereas the Greeks (specifically the Spartans and the Macedonians) had failed to do so: "Can any one be so indifferent or idle as not to care to know by what means, and under what kind of polity, almost the whole inhabited world was conquered and brought under the dominion of the single city of Rome, and that too within a period of not quite fifty-three years?" (1.1)

I'm less familiar with later historians, but this quote is of interest:

Varro's Antiquitates rerum divinarum [a history of Roman religion - ed] of 47 BCE opens with a dedication to Julius Caesar, exhorting him to remedy the neglect of the Roman religion. (Edwards, p. 49)

Livy, too, appears to have written his Ab urbe condita to turn his readers away from the degeneration of his times and towards the virtue of older times.

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Works mentioned:

• Herodotus, Histories
• Thucydides, A History of the Peloponnesian War
• Xenophon, Anabasis (also known as The March of the Ten Thousand or The March Up Country)
• Xenophon, Cyropaedia (also known as The Education of Cyrus the Great)
• Polybius, Histories
• Livy, Ab urbe condita (also known as History of Rome)

Source used:

• C. Edwards, Writing Rome: textual approaches to the city, Cambridge 1996

[Edit: spelling, typography]