Proponents of a Brexit claimed that the UK was paying a large sum of
money each week, which could be avoided by leaving the EU. John Oliver
claimed that the UK would still have nearly the same expenditure if
they weren't part of the EU.
True, the EU is giving a lot in terms of financial aid (agriculture, education, etc...) to the UK. At the same time, the UK is giving a lot of money to the EU, that is true. Without or with the EU, that won't change.
Proponents of a Brexit claimed that leaving the EU would free them
from many EU rules and regulations. John Oliver claimed that the UK
would still have to follow EU rules and regulations in order to do
trade with the EU market.
Also true; that is why the EU is working on a treaty with USA, Canada, and so on for their products, since a lot don't respect the EU norms. If the UK wants to exchange products with the EU, everything would have to follow EU regulations; same goes for fishing, etc.
So then, why the Brexit? There are several reasons; the first one is to pressure Brussels by showing that the people of the UK are not really satisfied with the current state of the EU (A lot of countries are not, in fact). A short victory for "Remain" would have given a lot of political strength to Mr Cameron in Brussels, which would have enabled him to negotiate a lot of advantages (economic, political) for the UK.
Inside the UK, the idea was to give political trust to Mr Cameron, since he was in a bad situation. Too bad, he has to resign; the bet was fine, but he lost.
UKIP lied to a lot of people who trusted the party until the end, and now the leader of this party is saying he can't keep his promises. That will maybe end up with a political mess in the UK right now.