How do I browser folders in Lollipop? How do I browse my Downloads folder on my Nexus 6? Or Pictures or Audio or any folder?

Right now, I do so by opening Gmail, and browsing possible attachments, which is silly but works.
Simplest of all, by downloading one of the many filemanagers from the Google Play Store or from your favorite Android app store. These would allow you to show files there. 
For files downloaded via Android´s built-in download tool, you can access the ´Downloads´ app on your Nexus as well. However, that will limit you only to starting an associated app for the download (if one is available).

One good example of a filemanager I use myself is FileCommander, which can be downloaded from the Google Playstore here