After accessing the Asset Store and viewing the asset you'd like to download, simply click the Download button. The button text may also be Buy Now if the asset has an associated cost.
If you are viewing the Unity Asset Store through the web interface, the Download button text may instead display as Open in Unity. Selecting this button will launch an instance of Unity and display the asset within the Asset Store window.
You may be prompted to create a free user account or sign-in if it is your first time purchasing from the Unity Asset Store.
Unity will then proceed with accepting your payment, if applicable.