Filtering a date field in views
Is the "Start Date" field a date field? Or is it a text (or some other type of field)?

If it's an actual date field, you can configure this behavior through the UI.

Add the field as a filter in the Filters section. Then you should see an Is between operator radio button. Select that and enter your range in the Value section. Screenshot:


You need to activate the date_views module to see the filters appearing in the Views module interface.