Suppose the exchange rate is 1.5 USD = 1 GBP. Suppose in New York there is a hamburger you can buy for $3 and in London there is a hamburger you can buy for 2 GBP. At this time the prices of the two hamburgers are effectively equal since 3 USD = 2 GBP. So you can say the price of either hamburger is 3 USD, or you can say it is 2 GBP; they are the same. If you were sitting in New York with $15 in your hand, you could buy five hamburgers, or you could go to London, exchange the $15 for 10 GBP (15/1.5 = 10), and still buy five hamburgers. Likewise if you were sitting in London with 10 GBP, you could buy five hamburgers or go to New York, trade your pounds for $15 (10*1.5 = 15), and still buy five hamburgers.
Now suppose that due to some event (such as the Brexit vote) the exchange rate shifts. Now 1.25 USD = 1 GBP. It is unlikely that hamburger vendors in both cities will immediately update their prices to reflect the change. Thus the hamburger in New York still costs 3 USD and the hamburger in London still costs 2 GBP. But because of the change in exchange rates, if you measure the prices in the other currency, you find that hamburger in New York now costs 2.40 GBP (3/1.25), while the hamburger in London now costs only $2.50 (2*1.25).
The prices of the hamburgers have now diverged. If you were sitting in New York with $15, you could buy five hamburgers -- or you could exchange your 15 USD for 12 GBP (15/1.25 = 12), and when you went to London, you would now be able to buy six hamburgers, because each one costs only 2 GBP. Likewise if you were in London with 10 GBP, you could buy five hamburgers, but if you exchanged your 10 GBP for $12.50 (10*1.25 = 12.5), when you got to New York you would no longer be able to afford five hamburgers, because they cost $3 each.
This is what is meant by saying that a given exchange rate makes things cheaper or more expensive in one country versus another. It means if you were to exchange your money for the other currency, it would buy more (or less) of the same goods in the other country. In the longer term, the fluctuation of exchange rates may be intertwined with other economic changes that could cause prices to shift for other reasons, so the "hamburger arbitrage" opportunity might not last. But from the perspective of a traveler going from one country to the other, it just means there are at least some goods available at prices that are not commensurate when converted to a common currency.