I hadn't heard 「さつき」used in ages and I didn't remember what it meant until he said it was "May."
As you may or may not know, Japan used to follow a lunar calendar (hence why months are represented by 「月」, the character for "moon"). I'm not sure exactly when the names were changed, but sometime after Japan's switch to the solar (Julian) calendar, it also switched what it called each particular month. The old names are still used from time to time, though, as in this expression.
皐月 (五月、さつき) came about by a shortening of an even older name for the month. Because it was during the time when rice was planted, it was originally called 早苗月 (さなえつき). Over time, people shortened that by cutting out characters, and it became さつき.
It seems that 五月晴れ (さつきばれ) is used most often by newscasters or weather forecasters during nice, sunny days in May. According to my research, this usage is/was actually in error. When Japan switched to the solar calendar, all of the months shifted over by one. So even though さつき refers to May, it actually corresponds to what would be June today. This has either been largely ignored or forgotten, though, and the term 五月晴れ was at one point amended in Japanese dictionaries to note that it can in present day refer to May, despite its original use in June.