I'm Tom Lehmann, a professional game designer, with over 50 published gaming products, including Dice Realms, a game based primarily on changeable dice, and the Orb Game in the Rivalry expansion to Roll for the Galaxy.
Have you ever rolled a die in a game and wished you could change the odds of a given face coming up? Changeable dice let players do that.
Have you ever wanted special powers in a game that only occur some of the time?Changeable dice let players do that in a very elegant manner.
Have you ever wanted to rapidly prototype a dice game with custom faces?Changeable dice let game designers do that.

Let's explore these three different uses:
Dice Realms is an example of a game which is just about the changeable dice themselves. Players each start with two dice and can acquire more during play from a common pool.
The dice faces can be improved via upgrades into better versions of the starting faces or swapped out for completely different faces from a set of 5 face types on offer, which differ from game to game.
Do you improve your die faces to earn more victory points and upgrades or do you put lots of money faces on them to be able to buy more dice and then do more things every round? Gaining a dice advantage over your rivals is powerful, but time and the victory chip pool is ticking away...
With two starting dice, you can specialize your dice by putting two or three of the same faces on a dice, dramatically altering the odds of what you will rolll. Do you specialize for money, upgrades, victory points, grain, defense, etc.?
Turns are straight-forward: roll your dice; collect grain, coins, and upgrades; possibly spend coins and grain to buy more dice; and then spend upgrades to improve your die faces. Dice Realms is all about the changeable dice themselves.

In the Roll for the Galaxy: Rivalry Orb game, by contrast, each player has a single Orb die which they can assign workers to improve via the Research action.
Orb die face improvements open up new technologies which interact with the core game actions of Roll for the Galaxy: explore, develop, settle, produce, and ship. Which technology do you boost? How much do you invest in buying better die faces to boost that tech? You can invest a little or a lot, but investing a lot means your workers are doing other useful things... 
When you want improvements to not be all or nothing, but to occur some of the time and at varying intensities, representing them as changeable faces on a die works very well. Here, the Orb die and its changeable faces are a means to an end, not the end itself.

Also in the Rivalry expansion is my co-designer's Deal game, created by Wei-Hwa Huang. The Deal game uses 7 custom dice with over 25 different faces among them. One issue Wei-Hwa faced when designing the Deal game was which given die faces should be split among several dice and which should be paired up on a single die?
Changeble dice meant that Wei-Hwa could easily swap his various die faces among the 7 Deal dice and try out different arrangements of faces among them quickly. After coming up with the best arrangement, one production option was to then make custom dice with those faces. Here, changeable dice are being used as a quick prototyping and game development tool.
(In practice, as we were already going with changeable faces for the Orb Game, we also used them for the Deal game in actual production, even those no changing of faces actually ever occurs in the Deal game.)

Changeable game faces allow for quick prototyping and open up new areas in game design: the ability to improve die faces and the ability to concentrate multiple copies of faces to alter the odds in your favor. Enjoy!

further reading
Tom Lehmann on BoardGameGeek
Changeable Dice Product Page