Apple's reasoning for the throttling was understandable, as the company explained in an open letter to customers around this time last year:
Device use also affects the performance of a battery over its lifespan. For example, leaving or charging a battery in a hot environment can cause a battery to age faster. These are characteristics of battery chemistry, common to lithium-ion batteries across the industry.
A chemically aged battery also becomes less capable of delivering peak energy loads, especially in a low state of charge, which may result in a device unexpectedly shutting itself down in some situations.
It should go without saying that we think sudden, unexpected shutdowns are unacceptable. We don’t want any of our users to lose a call, miss taking a picture or have any other part of their iPhone experience interrupted if we can avoid it.
The problem for Apple is that it didn't explain all of this to customers and instead secretly implemented controls with iOS 10.2.1 that throttled processor performance in order to preserve battery charge and prevent "unexpected shutdowns". It wasn't until third-party testing finally uncovered the practice that Apple responded and said that it would be more transparent (staring with IOS 11.3).
Apple's $29 battery replacement program was instituted to give customers with out-of-warranty iPhones the ability to get a get a fresh battery, no questions asked, at a reduced cost. An out-of-warranty battery replacement for iPhones usually runs $79, so a $50 discount is pretty good. The only problem is that you have roughly a week to get the service performed on your iPhone before program ends.
If your iPhone is still under warranty, and fails Apple's battery tests, you'll of course get your replacement for free. But if your battery fails and is out of warranty, you'll pay the $29. Even if your battery is humming along with 95% maximum capacity, or even 88% capacity (both of which are within spec), Apple will still replace the battery for $29 if you request it.
If you want to have your battery replaced, your best bet is to schedule an appointment with a local Apple Store Genius Bar (where they can usually get in you in and out within 45 minutes to an hour) or show up without an appointment where you could end up waiting for 2 hours or more.