Auto-update tools in Me is on by default: when a tool has a new version, it's installed in the background for you.
If a new version would access new websites, use new credentials or add background tasks, it isn't updated automatically. Instead, you're asked first: the number of available updates appears next to the title on the Tools page. Tap it to review them one by one. The top of each panel says "New in this version: …", and the tool only updates when you tap "Update".
With auto-update off, the number of available updates also appears next to the title on the Tools page; tap it to update them all. Tools with a new version also show a small red dot.
An update only replaces the tool itself (widget design and how it gets data). Your widgets and their parameters are kept. If a new version no longer offers a widget, that widget is removed from your dashboard.