Ok, going to try to make a long, complicated, and contentious story as simple as possible:

In fact, they were not so much drawn after WWI as during. The process would begin shortly after the outbreak of the war. The dividing of enemy territories not yet conquered between allies was common at the time, and the British and French had a deep interest in the region for a variety of reasons. The central British concern was the protection of the Suez canal and the continued British domination of Egypt, which was a de facto British colony as of 1882. The primary French goal was to stymie British control of the region (Britain and France, while allies, had a very adversarial relationship). Both powers, as was a common colonial tactic for the time, would claim a status of "protectors of minorities", the French claiming themselves the protectors of the Christians of the region, and the British claiming themselves the protectors of Druze and Jewish populations. In addition, the oil fields of modern day Iraq were known to exist, at least later in the war, and would become another important strategic goal.

As part of the ongoing war against the central powers, the British in particular would make a series of three agreements which would decide the fate of the regions post-Ottoman boundaries (keep in mind that this is a brief outline, as entire books have been written about these agreements).

A British diplomat by the name of Mark Sykes, and his French counterpart Georges Picot would draw the first drafts of the Sykes-Picot agreement in 1915, and it would be concluded in 1916. The agreement gave Britain control of modern day Palestine, Jordan, and southern Iraq, and the French would have control of modern day Mosul, Syria and Lebanon. The Russians were party to the agreement as they were key members of the Entente alliance, and would have received part of modern day Turkey, though this would never happen.

At the same time, the British were attempting to coax the Arabs into rising up against the Ottoman Empire to help the war effort. The British High Commissioner in Egypt, Henry McMahon, maintained a correspondence with the Sherif of Mecca, Hussein, seen by many in the British Cairo office as the best potential leader for the Arabs due to his role as Sherif of the holy Islamic places. In 1915, McMahon would state that the British were willing to recognize an independent Arab state consisting of modern day Palestine, Jordan, Iraq, Syria, and Lebanon, with a few exceptions:

"The districts of Mersina and Alexandretta, and portions of Syria lying to the west of the districts of Damascus, Homs, Hama and Aleppo, cannot be said to be purely Arab, and must on that account be excepted from the proposed limits and boundaries. With the above modification and without prejudice to our existing treaties concluded with Arab Chiefs, we accept these limits and boundaries, and in regard to the territories therein in which Great Britain is free to act without detriment to interests of her ally France, I am empowered in the name of the Government of Great Britain to give the following assurance and make the following reply to your letter: Subject to the above modifications, Great Britain is prepared to recognize and support the independence of the Arabs within the territories in the limits and boundaries proposed by the Sherif of Mecca."

The Sherif would rise in revolt in June 1916, and is worth an entry of its own; T.E. Lawrence "of Arabia", as well as many other British officers and diplomats, would arm, train, and help lead the Arabs against the Ottomans for the duration of the war.

Throughout the war the issue of Zionism would become increasingly important in Britain. Influential Zionists, such as Chaim Weizmann, as well as British officials in London, would continually lobby the British government to support the creation of a Jewish state. One reason given for this would be the needed global Jewish population's support in the war; the NILI spy network in Palestine against the Turks set up by Aaron Aaronsohn would be used as an example of international Jews aiding the war effort. In 1917 Foreign Secretary Arthur Balfour wrote a declaration to British Zionist leader Baron Rothschild stating:

His Majesty's government view with favour the establishment in Palestine of a national home for the Jewish people, and will use their best endeavours to facilitate the achievement of this object, it being clearly understood that nothing shall be done which may prejudice the civil and religious rights of existing non-Jewish communities in Palestine, or the rights and political status enjoyed by Jews in any other country."

The Sykes Picot Agreement would be revealed when the Russians would pull out of the war during the Bolshevik revolution, as the Bolsheviks were decidedly anti-colonialist. This would cause extreme distress among the leaders of the Arab revolt as well as the global Zionist community, both of whom would be given assurances that their claims would be upheld. In September of 1918, the Arab revolt would reach Damascus and proclaim an independent government in the name of Sherif Hussein. The Arabs would declare a state in 1920, using the boundaries set forth in Hussein-McMahon, consisting of modern day Jordan, Syria, Palestine and Iraq.

The San Remo conference would be convened to discuss what to do about the situation. British PM Lloyd George secretly agreed to grant the French a mandate over Syria and Lebanon (an increasingly unpopular move, especially among the Arabs who refused to be under a French mandate), as long as the British were granted control of Iraq, and importantly the oil fields of Mosul. The states of the region would be officially solidified at this point: France was to receive mandatory (essentially semi direct control of a state until it is deemed "ready" for independence), control over Syria and Lebanon. Britain was to receive indirect control of Trans-Jordan(Jordan and the West Bank) and direct mandatory control of all of Iraq and Palestine. It was agreed that the Palestine mandate would accommodate for the immigration of Jews and be a "national home" for Jews, however the exact meaning of this was vague and to be determined by the British.

France would see this as justification for its claims in Syria, and would send in its army in 1920 to depose Sherif Hussein's son, Faisal, from his position as the King of Syria. Faisal would later be taken in by the British and made King of Iraq, Hussein's other son Abdullah would be made King of Jordan.

Thus, the borders of the Middle East would remain drawn as such until the creation of Israel and Israeli independence in 1948. That is a whole other long and complicated story, however, and I already feel as if I have brushed over and simplified so much important information in just the WWI and aftermath period already.

As for more info, there are a few sources which I think are excellent if you want more in depth information. "A Line in the Sand" by James Barr is an excellent book about the British French rivalry over the region, and how those lines were drawn in the first place, covering the late 1800s until the creation of the state of Israel. "Lawrence in Arabia" by Scott Anderson, while focusing mainly on the people who made these decisions on the ground, is an excellent look at the strategic concerns on all sides of the conflict and negotiating table. "Jerusalem 1913: the Origins of Arab Israeli Conflict" by Amy Marcus is a very good and reasonably unbiased account on how these developments would affect people on the ground, in Jerusalem in particular. And like I said, I have definitely brushed over and skimmed quite a bit, so if you'd like any clarification dont hesitate to ask!