Let stay on basic level, there are other ways to accomplish it, but with this you still can follow the logic.
Later on you can use other techniques....
You have your DOB field, which is of type 'date'.
Create a new field; monthDOB_ct.
It is a calculation, result text.
In the calculation window, in the upper right pane, you have all the functions.
Scroll down untill you find MonthName(date).
Double click and the function will appear in the calc box, with date highlighted.
Scroll down in the left pane, where all the fieldlabels are, untill you find your DOB field, double click.
( data ) will be replaced by ( DOB ).
Make sure to change the calculationresult to text in the lower left box before clcicking OK.
Back to your layout, put this new field on the layout.
This is the field to search on.
Go in to Find mode, give a month in this field and FileMaker will give you all the records with that month.
I never let users search on a data entry layout, I always redirect the user to a special searchlayout, where I put all the fields from the given table where it is allowed to search on, but that's another story.
Instead of monthName you can use Month( date ) instead, this will return the monthNumber instead of the name.
The way of working is the same.
If problems, feel free
)