How is the Labour Party, with 57 fewer seats, able to still be in power in government?
Did the Labour Party hang Parliament?
Each constituency elects a member of parliament. There are 650 seats up for grabs. To have a majority, a party needed to have at least 326 seats. Otherwise the election is hung (basically a draw for want of a better term).
Now the parties have a choice:
Labour doesn't have the option of forming a coalition government. Its seats plus the other, non-conservative party seats together don't add up to 326.
Labour could potentially (unlikely though) form a minority government.
To answer your question of why labour has a lot of power in parliament now, is that the conservatives can't pass bills on their own any more, and labour can influence individual bills by working with other parties.
Firstly, the Labour Party didn't hang the Parliament. It's called a hung parliament because no party won a majority of seats in the House of Commons. As there are 650 seats in total, a simple majority requires half that number + 1, which makes it 326.
The expression comes from the term "hung jury," when jurors cannot decide on a verdict and a retrial is needed.
The Labour Party can form a government if they can get a working majority either through forming a coalition or through a "confidence and supply" deal.
A coalition is more formal and that the smaller party usually get some ministerial positions in the government, for example the Conservative–Liberal Democrat coalition agreement in 2010 formed a cabinet consisting of ministers from both parties.
A "confidence and supply" deal's less formal and the government technically remains a minority one and the smaller party agrees to support any bills necessary for the government and in return the governments helps them to enact some of their policies.
A hung parliament is a result of no majority. There are 650 constituencies (geographical area that each individually vote for an MP) in the UK. In parliament, a majority is achieved when the largest party has more seats than all other parties put together, which requires 326 seats. The reason for this is that there are many more liberal parties in the UK Parliament than Conservative, and chances are that during motions that have to carry in parliament that need a majority conservatives will have a hard time passing laws if the Liberal Democrats, Green Party, SNP and Labour all vote against the Tories' motions. This is why it is important that to properly win an election a majority is needed.