There is not any direct link between the browser and Moneydance. The browser allows you to download the transaction file to you computer. You can use the Moneydance ' menu command File --> Import to load the file into Moneydance. or if you have file associations set double clicking on the file in the file browser application should do the same.
The purpose of the internal browser is to be able directly link to Moneydance as well as to be able to automate the task.