I was playing around with the Google SDK and realised that I could not browse the filesystem/external storage, etc. I feel extremely silly asking this, but how do I browse the SD card?
No, there isn't a stock file manager. Viewing the SD card's directory layout is a very desktop-PC, old-fashioned way to do things, and most phone users don't have a need for it, so there isn't a default file manager in vanilla Android. Media files on the SD card are automatically scanned by the OS so they can be found by the apps that support them, so just use whatever app you would use to open the file.
For example, if you're looking for image files, go into the Gallery, where all images (whether they're files on the SD card, automatically synced from the web, or come from somewhere else) can be found in one place.
If you're desperate to see the directory layout itself, you can find many file manager apps on Google Play.
I know this is late, but you actually can navigate your files from your web browser. Enter
file:///sdcard as your URL and you should be able to navigate and access your files.