To answer the second question first, Walter Rothschild was from the British branch of the family (which originated in Germany I believe, hence the name which translates to "Red Shield"), and was one of the foremost Jewish men in Britain at the time, being the 2nd Baron Rothschild and a wealthy banker in his own right.

As to what was in it for Britain, this is more complicated and historians still argue over Balfour's exact motivations. James Gelvin (The Israel-Palestine Conflict: One Hundred Years of War) argues that Balfour and others were ironically somewhat anti-semitic in their beliefs that Jewish influences had the ear of President Wilson, and sought to attract their favour, and likewise looked at the then-ongoing Russian Revolution as a Jewish-led thing. Plus of course the money controlled by Jewish financial interests.

Others point to the British Government wanting to make the inhabitants of that area particularly beholden to them for when the post-war wrangling with France over who got what in the Middle East broke out.

David Fromkin covered it in a Peace to End All Peace as part of a race to establish British influence in Palestine before the French could, as both sides were keen on grabbing the territory. The United Kingdom had the confidence of Zionist leaders, and the French really didn't. Furthermore you did have important British officials like Herbert Samuel who were sympathetic to Zionism, although not necessarily active supporters, which made a Jewish home a natural way to gain Palestine away from French hands.

This matters because under Sykes-Picot, Palestine was an International Zone, so Britain was keen on securing it under their influence before the French could do so ( and indeed the French were interested).