I'm an American so I'm not super knowledgeable about U.K. party politics. What is a party member specifically and why are the numbers so low compared to the population as a whole?
I can't imagine a far left guy like Corbyn getting elected if everybody who votes Labour had a say.
A party member can vote in party elections to determine who their local candidate for that party will be, or for party positions like leader or chairman. I guess the equivalent is something like registering as a Democrat or a Republican in the United States. However, you have to pay a small amount to be a party member. The US doesn't do this because primaries are funded by the states and so the Democrats and Republicans don't have to meet the costs of organizing them. For internal matters like primary elections, UK parties have to meet their own costs, so there's a membership fee - e.g., I pay a pound every month to be a member of the Labour Party (about $1.30).
Parties used to be internally quite strongly democratic, especially the Labour Party, where much of the policy was decided by the votes of party members at the party conference. But politicians decided to 'professionalize' parties and reduce the input of members, in order to dump policies that appealed to the members but not the electorate. That gave less and less reason to join political parties, and there's still the fee attached, so membership slumped heavily:
(note that this is a little misleading - the Labour Party allows anyone to vote who is a trade union member, broadly speaking, so it had a huge voting base that didn't feel the need to become a member; it was a mass participation party, and much more so than the Conservatives, which this graph doesn't really show. However, trade union membership has also plummeted, following the same trend)
In 1950, participation in the party process was about a fifth of the population; or roughly the same proportion that participate in the primary process in America. Now, it's a fraction of the population - nobody really cares enough to bother.
The Labour Party's electoral system for the leader used to be an electoral college, where MPs got a third of the vote, members a third of the vote, and affiliates (trade unionists) a third of the vote - sort of like the primary system in the US where sitting Democratic politicians as superdelegates control an outsize amount of the vote compared to people participating in democratic primaries. Ed Miliband, perceived as a leftwing candidate at the time, narrowly beat David Miliband, perceived as the rightwing candidate, thanks to the vote of affiliates. The right of the Labour Party took this quite poorly, and put huge pressure on him to reduce trade union input. He did this by switching to one man one vote.
This had a backfire effect, because while it took power away from the trade unions towards the members, it also took it away from the MPs towards the members. Corbyn managed to get an influx of new and enthused members that shocked the Labour Party by voting him in, as the most leftwing leader of the Labour Party in relative terms since, uh, Lansbury (1932-35). This would be sort of the American equivalent of, um, the Democrats abolishing superdelegates and caucuses because a Blue Dog faction felt that superdelegates were favouring leftist candidates like Obama (e.g. Ed Miliband was basically a centrist), and instead having someone like Sanders win.
It's a very rough analogy, because American parties have very different blocs to British ones and I don't think the Democrats are at all comparable to the Labour Party, but as 'translations' go, it's about as good as you'll get.
