I'm not a professional historian, but I am a student whose study focus is the Middle East, so I feel like I can answer this one.

To understand how the Mandate of Palestine and the Mandate system in general came about, you need to go back to World War I. The Ottomans joined the war as part of the Central Powers (Germany and Austria being the other two core members), against the entente powers (Britain, France, and Russia). Starting in 1915, the entente powers started to discuss how to divide up the Ottoman Empire once the war ended. Russia wanted the Turkish Straits to have access to a warm water port, France wanted Syria and Lebanon due in part to its historic role as protectors of the Maronite Christians in the area, and Britain wanted to maintain routes to India and preserve economic interests. The entente powers started to write down these interests in secret agreements, such as the Constantinople Agreement (Russia gets Turkey, France gets Syria, Britain gets Persia) and the Sykes-Picot Agreement (division of Ottoman territory into areas of French and British control). Britain not only signed agreements with the entente powers, but with local and nationalist groups as well. For instance, Hussein, Sherif of Mecca, was promised a vague Arab "state or states" in exchange for his leading a rebellion against the Ottomans. Another agreement which you point out is the Balfour Declaration, where Britain said it would support the creation of a "national home" in Palestine for the Jews (the exact reasons for this declaration are disputed).

All these declarations were contradictory and ambiguous. Just looking at Palestine, one agreement (Sykes-Picot) promised Palestine to the French as part of Syria, while the Russians viewed the same agreement as saying that Palestine was just defined as the area around Jerusalem, which would be an international zone. Sherif Hussein, meanwhile, interpreted Palestine as part of the promised Arab state(s). Further complicating matters were regime change in Russia (the new regime was atheist and had no concerns about Orthodox access to Jerusalem, so the proposed international zone which was proposed to address this point was suddenly unnecessary) and US pressure to follow Woodrow Wilson's Fourteen Points, which included as #12 the expectation that "nationalities which are now under Turkish [Ottoman] rule should be assured... an absolutely unmolested opportunity of autonomous development." According to the nationalist leaders of the Middle East, this meant that they needed to be represented in post-war negotiations.

To resolve these and other international disputes, the entente powers created the League of Nations. The charter of the League solved the problem of Ottoman territories by splitting the land up for British and French control into so-called mandates. Britain got control of what would become Mandatory Palestine among other territories in the region (the British mandate included what are now Israel and the Palestinian Territories, Jordan, and Iraq). The mandate system was created as a compromise between France, Britain, and the US, and the League of Nations specified that Britain and France were to help their mandates eventually become independent states. Thus, Mandatory Palestine was born.

(I'll continue this in a reply, worried about character count). Source: "The Modern Middle East: A History" by James Gelvin