Why does the gold standard require countries to keep exchange rates fixed?

This is required just due to the simple definition of what gold standard is. For a monetary arrangement to qualify as gold standard money has to be convertible into gold. Hence central bank or some other government body must be always prepared to convert notes into gold. That is part of both the dictionary and economic definition of how it works.

To see why this requires fixed exchange rate lets say that this gold conversion is set in US such that $$1$ converts into $1$ gram of gold. Now suppose EU would set its conversion rate to $1e = 2g$ of gold. If the exchange rate would be anything else than $E=frac{$}{e}=2/1$ the gold standard would eventually have to collapse in long run.

For example if exchange rate $E$ would be $1$ (i.e. $1EUR=1USD$) then everyone would take their dollars convert them to euros demand gold from European Central Bank to the point when either it would left without any gold to support the gold standard or it would change the exchange rate. If the exchange rate would be higher than $E=2$ opposite would happen. Consequently, under gold standard maintaining fixed exchange rate is necessary otherwise it cannot survive due to the fact that under gold standard notes must be convertible into gold.