Before you import any assets, you need to choose either to Append or Link the assets. So, what are the differences?
Appending works by making a copy of the rig from the source to the current blend file, while linking works by creating a reference to the data in the source file. That way if you make modifications to the source file, the changes will be reflected in the referencing file.
Note that if you choose to Link certain assets like a rig, you need to Library Override it to make it editable. To do that, select the linked collection in the outliner, right-click, and select Library Override -> Make -> Selected & Content.
Select the collection, right-click, select Library Override > Make > Selected & Content
Importing Method
Here's how to import assets from the Asset Browser or using the manual method:
Open the Asset Browser in your workspace.
Open the Asset Browser
Select the EndRig asset library.
Select the EndRig Asset Library
Optionally, change the Import Method (Append, Link, or follow your asset library preferences).
Changing the Import Method
Click and drag any assets you want into the viewport.
Click and drag assets into the viewport to import it
In the Operator panel, uncheck the Instance option.
Uncheck the Instance option in the Operator panel
Go to File -> Append/Link. A window then should pop up.
Go to File > Append to open the file browser
Look for the blend file you want. Double-click it and it should show all the files inside. In this case, I want to open the rig file to get the rig asset.
Find the blend file and double-click to open the file
Here you can access the blend file contents. Since I'll be importing the rig, double-click on the Collection folder, find and select the rig main collection, and then select Append/Link.
Double click on the collection folder to see all collections
Select the rig main collection then select Append/Link