From Axios Macro:
In a currency war, e.g., between the US and EU, the US reduces interest rates, which makes investing in the US less attractive, which means there are fewer foreign investors who want to sell euros to buy dollars to invest in the US.
[We look at this from the point of view of dollar demand and supply, so when you see the word "buy" think "demand"].
This reduction in the demand for dollars reduces the price of a dollar, which makes US exports more attractive to foreign consumers which helps US producers, but hurts foreign producers. To protect their own producers by making the euro weaker, the EU will reduce rates in response.
In a reverse currency war, the US raises interest rates to combat inflation, which makes the dollar stronger, which slows the US economy by hurting US producers as it makes US exports more expensive.
However, this also stimulates the EU economy by making EU exports to the US look more attractive to US consumers. To avoid "importing" US inflation, the EU will raise their own interest rates.