You can connect assets to other assets or nodes in the Hypergraph Connections editor by dragging connections.
Before you can do this, you must publish the output attributes of an internal node to the asset.
To create a new connection from an asset to an external node
Maya draws a line from the node to the mouse cursor.
A list of input attributes is listed when you hold the mouse button.
Maya creates a connection between the two attributes.
If the list appears empty when you click the bottom-right corner of the asset, it is because you did not publish an output attribute.
You can also middle-drag the asset onto an external node and select an attribute to connect to from the list that appears. This opens the Connection Editor where you can setup multiple connections at once.