I'm always surprised when I work on a software team that doesn't use the product they're building. It's a sign that your team isn't invested in the product. More importantly, it's a sign that your team isn't as invested in the customers.
Teams who dogfood their own software make better products. That sounds anecdotal, which it may be, but that's what I've seen in practice. Why is software better when those that make it use it?
Your team is always in the ecosystem
If you aren't immersed in the ecosystem that your app lives in, you won't ever truly understand how your app should work. You won't understand what metaphors are the right ones and which ones will feel foreign or fake. You can be sure your customers will notice if you use an iOS metaphor on an Android phone or a Windows metaphor on a Mac. Until you are immersed in the environment your customers are in you won't really understand your customers as well as you could.
Your team is your customer base
Relying on your own software in your day to day life builds empathy for your customers because you are part of the customer base. You'll find the workflows that need simplifying. You'll understand which features are missing. You'll cringe and curse every crash.... just like your customers do.
Your team feels any added friction
When your team is using your app you discover added friction before your customers do. For instance, if you make a change that's not backwards compatible, your team will find it first because they're already using the app and that feature. Add additional, and unnecessary steps to some workflow, your team more likely to find it than a paying customer.
Long and short, you find the mess in your app before anyone else does. When the app gets to your customers, they're delighted with how intuitive your app feels.