A couple of things:
1. you'll need another table with year/months in it - suggest as 201501, 201502 etc - reason is you need to report a figure for a month even if nothing started that month. and you need the year/month to differentiate between January this year and Januarys in previous years.
2. I presume that a service started in January and reported in February and hasn't finished still need to be reported
So with this in mind and assuming your YearMonth table is populated with say 201501 through to 201512 try the following query I've assumed your field names, and guessed at a table name.
Code:
SELECT UserID, right(YearMonth.yrMth,2) as Month,count(*) as services_count
FROM myTable, YearMonth
WHERE format(date_start,"yyyymm")>=yrMth AND (date_end is null OR format(date_end,"yyyymm")<=yrMth)
GROUP BY UserID, right(YearMonth.yrMth,2)