You're right; when it is running in the browser, isomorphic-git does not have access to the operating system's file system. Instead, it uses BrowserFS [1], which emulates a file system abstraction on top of arbitrary storage backends. It supports IndexedDB, localStorage, and Dropbox, among others. (Disclaimer: I am the author of BrowserFS.)
[1] https://github.com/jvilk/BrowserFS