Well, a few reasons, many of which changed over the course of the war.

In the initial stages of the war the British concern was threefold,

1. the Ottoman threat to the Suez Canal, primarily in the form of a large (and failed) raid in early 1915. Obviously control of the canal was of vital strategic interest to British empire and to the entire war effort. While this raid was repulsed, the British troops in Egypt failed to press on into Palestine for several years resulting in stalemate that wasn't cracked until the Battle of Beersheba and Third Battle of Gaza in late 1917. I'll come back to this front later.

2. Separately from the Suez/Sinai/Palestine front, there was the Dardanelles debacle. So-called "Easterners" in the British government, which in this instance were led by Churchill, sought to identify areas outside of the Western Front (with its churning, unbearable slaughter) where a swifter victory might be able to be achieved. Churchill's original plan was basically to throw a portion of the Mediterranean fleet through the Ottoman defenses in the Dardanelles, into the Sea of Marmara and then engage in a sort of gunboat diplomacy with the Ottoman government in Constantinople. Unfortunately several of the ships that attempted to force the straits struck mines and sank leading the navy to engage in a more cautious policy and, while the blame for this is a topic of quite a bit of debate, the end result was mission creep, the landing of a large land-based invasion force and the Gallipoli campaign which was more or less an unmitigated disaster with high losses and nothing to show for it. The British would eventually cut their losses and evacuate from the Gallipoli peninsula entirely six months later.

3. Under the leadership of Churchill as First Lord of the Admiralty, the British Navy had been converted from coal to oil by the outbreak of the war. The entirety of the supply of this oil came from the Anglo-Persian oil company's wells in Persia (now Iran) and were shipped through the Ottoman-held port of Basra, then the only viable deep-water port in the Persian Gulf. As a result a large British invasion force landed at Basra quite literally on the same day that the Ottomans declared war on November the 11th, 1914. Encountering little resistance at first, a British expeditionary force attempted to drive north into Mespotamia only to be cut-off, surrounded, besieged and eventually forced to surrender in the Siege of Kut in early 1916. The British still held Basra, the attempted reliefs of Kut, the surrender itself and the casualties suffered in the appalling conditions of the siege meant that quite a lot in the way of blood, treasure and prestige had been sunk into Iraq by 1916.

So In the first stage of the war you had these three vital concerns, 1. the defense of the canal, 2. the chance to knock the Ottomans out of the war quickly. 3. Oil. Over time the first point became less of an issue and the second was clearly not going to happen. The third was secure so long as Basra under control. Now we're getting a bit closer to Lawrence, but first, war aims.

Very early on in the war (and indeed, some had speculated to this end before the conflict broke out) that in the event of an Ottoman defeat that British policy over the past century of maintaining Ottoman territorial integrity would have to be reversed and that the empire would have to be broken up. Uniquely among the theatres of conflict the Ottoman Empire's territories were going to be doled out as prizes. British war aims were first elucidated to this effect by the DeBunsen commision in 1915.

At the same time, the British were also deeply concerned about the fact that the Ottomans had declared the First World War to be a holy jihad. While the Ottoman Sultan was the caliph of Sunni Islam the British Empire's possessions Greater India and in East and North Africa meant that Britain had by far the world's largest population of Muslims. Prior to this, Sherif Hussein of Mecca had made overtures to Horatio Kitchener while he was in charge in Egypt about the possibility of a British-backed Sharifian revolt in the Hejaz which were basically ignored. Now, both the Ottomans and the British were to seek the Sharif's support because his religious credibility and standing would have a huge impact on the viability of the Ottoman Jihad and his backing of the British in an open revolt could theoretically deliver the entirety of the Ottoman Arab territories to the British with practically the stroke of a pen by a mass revolt, or so the British thought.

The result of this was the Hussein-McMahon correspondences, named for the Sharif and the British High Commissioner in Egypt who negotiated the agreement. In exchange for joining the British side in a revolt against the Ottomans the British agreed to recognize an independent Hashemite-led Arab kingdom in the Arab territories of the Ottoman empire, with exceptions carved out for British interests in Basra and the Persian Gulf (i.e. Oil, and Indian trade) and French interests as described as being "west of Aleppo" basically conforming to modern Lebanon with its large Maronite Catholic population.

Agreement was reached on these terms and the great Arab Revolt that Lawrence was so critical in was launched in 1916.

Separately to this, the British were also in secret negotiations with both the French and the Russians over who would get what in the event that the Triple Entente won the war. You can see the map they produced here: https://en.wikipedia.org/wiki/Sykes%E2%80%93Picot_Agreement#/media/File:MPK1-426_Sykes_Picot_Agreement_Map_signed_8_May_1916.jpg

Not shown on the map are the extensive Russian promises which were to include pretty much the entire extended Caucuses range north-east of the French territories down into eastern Anatolia as well as Russian control of Constantinople. These agreements were also concluded in 1916 around the same time as the Arab revolt was launched.

Neither the Arabs nor the other members of the entente were fully aware of the extent of British promises, or of the results of the DeBunsen Commission over Britain's own end-goal aims.

This was the picture when Lawrence comes on the scene. While the Arab revolt failed to produce the region-wide uprising that the British had hoped it did tie up huge numbers of Ottoman troops in exchange for only small arms, money and a handful of British liaisons like Lawrence. Lawrence was also able to knock out a good portion of the Ottoman rolling stock and train-engines without which it would be impossible to supply their armies.

Despite the serious setbacks up to 1916 in the various Ottoman fronts in Mesopotamia, Palestine, and Gallipoli, after this Ottoman resistance more or less collapsed after the Battle of Beersheba and it became a race to take as much territory as possible. These victories were great for morale as well as with the "Christian Conquest of Jerusalem" for the first time since the Crusades.

In comparison with the Western Front, however, the numbers involved were not enormous, probably around half a million men in Mesopotamia and Palestine altogether with many of those being colonial forces for India and relatively low casualties. This compared with the 5,000,000+ who served with the BEF on the Western Front and the millions of casualties there.

Also worth noting is that, in effect, the Turks outfought the Germans and the Austro-Hungarians. Although they officially signed an armistice with the entente in 1918, the Ottoman troops in Medina held out against the Arabs until 1919, and the Turkish War of Independence basically saw Turkey repel the British, the French and everyone else from their modern territorial borders and to re-negotiate large portions of the Treaty of Sevres which would have partitioned pretty much the entirety of what is now modern Turkey as you can see from this map here: https://en.wikipedia.org/wiki/Treaty_of_S%C3%A8vres#/media/File:TreatyOfSevres_(corrected).PNG

In the end the efforts basically paid off, for the British and French at least. British "betrayal" to the Arabs over Sykes-Picot is a heated topic to this day among Arabs, especially, and those who think that imperialism caused many of the regions problems (which I would dispute, as I've discussed elsewhere in this sub). These commitments (with the exception of the Gallipoli campaign) were not an undue burden on the overall war effort and as a direct result of the British contributions they were able to agree to much better terms than what had been agreed in Sykes-Picot and instead secured pretty much the entirety of their aims as defined by the DeBunsen commission.

The wisdom of those aims and how imperial policy played out is obviously a separate and much lengthier question.

Source wise: The two I always point to on this sub are D.K. Fieldhouse's Western Imperialism in the Middle East and Eugene Rogan's The Arabs: A History. On Gallipoli, it's hardly objective but in terms stated intentions and the importance for our purposes here Churchill gives an extensive and self-serving account of his thoughts and record in the relevant portions of The World Crisis. There's a good deal of primary source material on some of the negotiations I've mentioned, including the Sykes-Picot map I linked to as well as the translated texts of the Hussein-McMahon correspondences. It's definitely pop-history I've been reading Scott Anderson's Lawrence In Arabia which certainly provides some interesting and focused context for Lawrence's actions in the Arab revolt and that certainly makes a good companion to Seven Pillars itself.