For anyone finding this post through Google (as I did) here's the correct formula for cell F5 in the above example:
=SUMPRODUCT((MONTH(Sheet1!$A$1:$A$50)=MONTH(DATEVALUE(E5&" 1")))*(Sheet1!$A$1:$A$50<>""))
Formula assumes a list of dates in Sheet1!A1:A50 and a month name or abbr ("April" or "Apr") in cell E5.