John Rawls is probably one of the most important recent liberal political theorists, but it seems that few people understand what his arguments in favor of liberalism were.
How would you describe Rawls liberalism to a layperson (i.e. not another political theorist) and what were the reasons that he believed such liberalism to be valuable
The meaning of the word "liberalism" in the English language is distorted by American politics and vernacular usage. I am assuming you mean economic deregulation. He was not totally in favor of it (not as much as anarcho-capitalists for instance), since it can impede social justice. Here is an easy way to explain his views on the matter, without any curves (the original text has some to illustrate his points).
A. Picture a society starting at a very egalistic repartition of revenue. Everyone has the same amount of wealth.
B. If you allow some moderate income differences, more resources are produced, and even the poorest can profit from these differences. This is desirable according to the second principle of justice (https://en.wikipedia.org/wiki/A_Theory_of_Justice#The_Second_Principle_of_Justice ).
C. If you allow even more income differences, you will, past a certain point :
reduce the amount of wealth owned by the poorest members of the society, which is not desirable according to the second principle.
reduce fairness of opportunity, not desirable according to the second principle.
Rawls aims to reach the middle point B. A liberal economy with measures such as progressive taxes in a closed economy is meant to reach situation B.
Note that, as explicitly mentioned in his book after the first principle of justice (https://en.wikipedia.org/wiki/A_Theory_of_Justice#The_First_Principle_of_Justice ), shareholding is not considered a basic liberty by Rawls.
I think the confusion comes from the fact that Rawls views led him to support many positions from the USA Democratic party, a.k.a. the liberals. This nickname comes from unrelated social issues (secularism, gay rights, ...), not economic or politically classic liberalism. In the 2016 election, economic liberalism in its original meaning was represented by Gary Johnson.
Following comment : In case the question was about political liberalism (and you can forget what is written above).
Equality before the law, secularism, freedom of speech, freedom of religion and gender equality, are all very clearly supported by the veil of ignorance argument. Quite simply, you can introducing it by asking someone "what if you were born in a religious minority/ethnic minority/political minority/discriminated gender/... ?"
However, the best arguments in favor of political liberalism were probably not given by Rawls, but by his predecessors. Rawls was in an environment where this was largely taken for granted (the first principle of justice suggests it). (In my understanding) his work in the theory of justice is not focused on this.
