How can I sum values of a specific category over a range of dates in Numbers?
You can most probably do this using, for example, the sumif() function (sumifs() if multiple criteria are to be applied). If you are looking for more help, you need to provide more details such as those I mentioned in my comment above.
 Update
Given the new information you have provided, below is one way I think the table you need can be created using the sumifs() function.

The picture above shows how to do it using the sumifs() function based on the samples of the input data table and the outcome table you have given. Pls note the absolute references in the formula, i.e. the &quot;$&quot; signs; they are necessary for the formula to work correctly when you copy &amp; paste it to other cells like I have done in the Analysis table above. Also note the beginning and the end dates per month added to the Analysis table.
If you have any difficulty in creating the formula in cell C4 in the Analysis table, you can recreate it by typing its following form (or copy &amp; pasting it) into an appropriate cell in your spreadsheet.
=SUMIFS(Data::$C$2:$C$9,Data::$A$2:$A$9,&quot;&gt;=&quot;&amp;C$1,Data::$A$2:$A$9,&quot;&lt;=&quot;&amp;C$2,Data::$D$2:$D$9,&quot;=&quot;&amp;$A4)

If your input data table is placed differently (the one in the example I gave, including the header row, starts from A1) or bigger than the one in the example I gave, you would need to change some or all of the cell references in the example formula provided here.
While you did not mention the Numbers version you are using, I expect you should be able to make good use of the example formula and the design I provided above as long as you are not using a very old version. I created the example sheets above using Numbers v6.1.