Asset widget module is included into Asset module package and is aimed to simplify Asset contribution and management.
All you need is to enable "Asset widget" module.
Widget will appear on content create/edit page if you have an appropriate permission and one of two next cases are true:
- Entity (node, taxonomy, etc) create/edit form contains Entity reference field with "Media Asset" target type.
- Entity create/edit form contains CKEditor with enabled asset plugin for at least one of the input formats.
The detailed contributor instructions are available here.