What does the period from July 23 to August 23 called "Rötmånad" in Sweden and "Mätäkuu" in Finland mean?

Rötmånad and Mätäkuu both mean rotting mouth. The period from July 23 to August 23 is the Dog Days of Summer and is listed as so by The Old Farmer's Almanac. It is called The Dog Days of Summer because this is when the highest temperatures of the year occur.
Rotting mouth refers to food spoiling due to these high temperatures. In hopes of escaping the heat, each year, Ancient Rome would sacrifice a brown dog. They would do so in hopes of making the bright star Sirius happy so that the star would bring relief from the high temperatures.