It is impossible to separate geopolitics and soccer, especially in international competitions. For the 1958 World Cup, in order explain how Wales qualified we first need to talk about what happened in the Africa/Asia region.
This was a time where Africa and Asia held joint World Cup Qualifiers due to the low number of teams, combined there were just 9 participants. One of these participants was Israel, and aside from China who got eliminated early by Indonesia, everyone else was a country of Arab majority. Political and religious tensions were high in the region at the time, and as a result everyone either withdraw or forfeited before facing the Israelis.
In theory, Israel should have qualified as the last team standing of the CAF/AFC Qualifiers, except there was one rule. FIFA requires every team to play a minimum of 1 match in order to qualify to the World Cup, so they had to come up with a solution. FIFA brought back from the dead all the teams who finished one position away from qualifying from all the other regions of the world and put them in a draw to see who would face Israel. With the exceptions of Uruguay who withdrew, and Italy and Northern Ireland who still had matches to play, the countries in the pot were Belgium, Bulgaria, Wales, Netherlands, Poland, Romania, Spain, Peru, Bolivia and Costa Rica.
Belgium was selected but refused to play. Next up was Wales who finally accepted the challenge, beat Israel both home and away, and qualified to their first World Cup.
More than anything this story shows how the World Cup grew in importance. Political tensions exist all of the planet to this day, but it would be unimaginable for so many countries to give up a chance at the World Cup to take a stand.