Hang on, don't liberals want small government? Why do they also want higher taxes then?

So, you need to bear in mind that there is a difference between classical and modern liberalism. Classical liberalism was the main type of liberalism in the late 18th century and throughout the 19th century. Modern liberalism became the main type in the early 20th century.
What you're saying about liberals preferring small government certainly applies to classical liberals. Philosophers like Jeremy Bentham, John Stuart Mill, John Locke and Adam Smith all argued that the state was necessary, but should have limited powers. They thought that this would prevent abuses of power, and help the economy flourish. However, particularly after the Great Depression, modern liberals such as John Maynard Keynes and Franklin D. Roosevelt began arguing that the government needs to take a stronger role in ensuring the well-being of citizens. Higher taxes were then seen by liberals as a means to fund an expanded social contract, in which the government provides welfare services to poorer citizens, and regulates companies to ensure economic growth and stability.