British Rule How?

The Ottoman Turks ruled over the area later known officially as Palestine. They considered it part of Syria. The Ottoman Empire was decaying. European Powers had taken to protecting minorities both out of sentiment and self-interest. Prussia (later Germany) became the de facto power for Protestants, France for Catholics, Russia for Greek Orthodox. When the British came along the only ones left were Jews. The British took to protecting them. Due to Jewish requests the British (in 1850) also extended their protection to the Samaritans and saved them from extermination at the hands of their Arab neighbors. So too in Russia and elsewhere the British, French, and Americans occasionally attempted to help the Jews when they were being persecuted.
 Along came World War-1 and the Turks joined Germany and the Austro-Hungarian Empire. Jewish settlement in Palestine had been on the increase but due to the war many of the Jews left due to persecution, famine, and expulsion. Some remained.
The Germans had been against Zionism but towards the end of the war they considered supporting it in order to sway public and Jewish opinion in the USA. Similar considerations held in England. Also in England a Restorationist Movement had existed for centuries. They believed in returning the Holy Land back to the Jews. Some of the Restorationists had, in part, missionary motivations but others wanted it simply because it was the right thing to do. The Restorationists also had adherents in the USA and France. [ US supporters included Professor George Bush who wrote a work on the subject in 1844. The Professor was a cousin of the ancestor of Presidents Bush]. Jews in Palestine and overseas helped the British quite well. There had been Jewish units in the British forces i.e. at first the Zion Mule Corps and after that the "Jewish Legion." These units were manned by Jews from all over the world. They had been created partly to wean Jews away from supporting Germany. In Russia Jews had been persecuted and the Germans at that time were viewed as potential liberators. The Jewish units in the British Army (in addition to their military value) were an answer to this. There was an understanding with the British that Jewish support for the Allies would lead to positive results concerning a future Jewish state. The British issued the Balfour Declaration in 1917 expressing British Government acceptance for the establishment of a Jewish Homeland in Palestine. The Turks were defeated and Palestine was conquered. Most of the Arabs had supported the Turks. A few had been allied to the British or to both sides. Those Arabs who had been allied with the British had been led by T. E. Lawrence i.e. "Lawrence of Arabia." He later became a strong supporter of the Zionist cause. From 1917 the British effectively ruled the Land. The French ruled in Syria and Lebanon. The League of Nations (precursor to the UN) gave a Mandate over the area to Britain. This confirmed the control they already had.

British Mandate for Palestine (legal instrument)
From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Br...)
The draft of the Mandate for Palestine was formally confirmed by the Council of the League of Nations on 24 July 1922, supplemented via the 16 September 1922 Trans-Jordan memorandum[2][3] and then came into effect on 29 September 1923,[2] following the ratification of the Treaty of Lausanne, with the United Kingdom as the administering mandatory.[4][5]

The Mandate given to Britain in effect gave legal sanction to the Balfour Declaration and charged British with fulfilling it.

British Mandate for Palestine (legal instrument)
 From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Br...)
 Whereas the Principal Allied Powers have also agreed that the Mandatory should be responsible for putting into effect the declaration originally made on November 2nd, 1917, by the Government of His Britannic Majesty, and adopted by the said Powers, in favour of the establishment in Palestine of a national home for the Jewish people, it being clearly understood that nothing should be done which might 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.[38]

And this how Britain came to control Palestine.
The British rule went through ups and downs. Many of the British personnel were anti-Jewish. British governments changed their polices and attitudes from time to time. The Arabs were often in revolt. Muslim unrest in India (which included Pakistan) and elsewhere threatened the British Empire. Anti-Semitism throughout the world was growing. Nazi propaganda had its effect and sympathizers. Some of the British worked openly on behalf of the Arabs. Others did not.
Despite all the above and everything else, on the whole the British furthered the Jewish cause. In hindsight the British helped create the State of Israel. They were in a de facto sense mostly on the Jewish side. This was not always evident and the British themselves were not necessarily always aware of it. Nevertheless such was the case when seen from an overall perspective.
 See the work,
"One Palestine, Complete: Jews and Arabs Under the British Mandate," by Tom Segev.