I highly suggest you read this post in order to get an idea of how the British Mandate developed, and how the state of Israel was created. It'll answer this question. Once you read it, if you'd like to know more or have questions, please let me know!

In short: the British Mandate didn't give it to the Jews, the UN Partition plan gave part of it to the Jews, the Arabs almost completely rejected it (most Jews accepted it), war ensued, the Jews won. The British Mandate released some documents saying they supported a Jewish state in Palestine, but gave no boundaries. Other discussions, like the McMahon Correspondences, are said to have also given a promise to create a Palestinian state (I think they did, but it's debated still). There are other claims, which I can get into regarding recent history (ie. Why the West Bank, why Gaza? These are based off 1967, not the British Mandate), but please let me know which part you want to know about before I elaborate there!