Do I need to reinstall apps to take advantage of app thinning? I don't notice any difference in app sizes in iOS 9 so I was wondering if app slicing doesn't kick in until next time I update apps or reinstall them?

I realize not all apps will support it, but I have so many that it should have shown a difference anyway.
You will need to re-install or upgrade existing apps to benefit. Apple is unlikely to pro-actively thin apps already installed on iOS devices.
The roll out may also be slowed by bugs and lack of developer support. For example in iOS 9.0, Apple disabled App Slicing due to a bug:

App Slicing Currently Unavailable
September 24, 2015
App slicing is currently unavailable for iOS 9 apps due to an issue affecting iCloud backups created from iOS 9 where some apps from the App Store would only restore to the same model of iOS device.
When a customer downloads your iOS 9 app, they will get the Universal version of your app, rather than the variant specific for their device type. TestFlight will continue to deliver variants for your internal testers. App slicing will be reenabled with a future software update. No action is needed by you at this time.