Here's how my setup looks here:
And this is how the formula should look one column over to the right after you've filled right:
Note that the only thing that has changed is B1 to C1. Everything else is the same. It's really important to have those $ "anchors" (the "Preserve Column") in the formula in B3 before you fill right to C3 etc.
If you're having trouble try copying this formula:
=SUMIF($Month,B1,$Amount)
then clicking in B3 and pasting it in.
BTW, if you're probably going to have more than one July (one in 2013, one in 2014, etc) you should format the Month column as Date & Time and also row 1 of the Budget table as Date & Time. Then make sure you type July 2013, August 2013 (i.e. month and year into each cell of each table where show each month). Though both month and year will be in the cell (actually Numbers will have month, day, and year: July 1, 2013, August 1, 2013...) you can tell Numbers to display just the month:
SG