You can easily export the list of assets under a category by following below method:
Go to Advanced Search> Manage Filters. Under Search Field select Category from the drop-down list, Conditions- Equals and Value- choose your desired category from the drop-down list. Click on Search button.
Search
results get displayed in the Browse widget. Change to List view. Click
on the Export icon. All the assets in the page gets downloaded in csv
format. Same method can be used to export list of assets under Project and Saved Filters.