Normally "to chill" means to lower the temperature. Here it means "to be less active". When a stock or currency is "hot", it is in high demand. When it is "cool", it is in less demand. This is also a pun on "subzero", where subzero means very chilled if you are talking about temperature.
The second part is missing a word. It should say "speculation ECB may ease rates subdues euro". Headline writers often leave out words to fit the space. Since we aren't using paper much anymore, there is no great reason to continue this practice.