Why is it better for an iPhone’s battery to NOT close down apps?
You’re not alone. People are just used to their desktop computer habits, and it’s understandable they carry these habits over to their iPhones and iPads. 

However, let me try and explain this using an analogy:

Imagine it’s a hot day and you’re outside gardening. You get thirsty, so you go inside to the kitchen, grab a large glass, put some ice in it, and fill it with water. You then drink half of it and empty the rest in the sink before going back outside. Not long later, you get thirsty again, so you go back inside to the kitchen, grab the same glass, put some ice in it, and fill it with water. Again you only drink half of it and empty the rest in the sink!

The above workflow just doesn’t really make sense. Why wouldn’t you take the glass outside with you? And, you’re not only wasting water by emptying it, but you’re spending more time and energy getting that water again.

Likewise, when you quit an app you’re actually using some battery power in the process of doing so (e.g. clearing it from RAM, etc) and then again later when you have to launch it again and load it back into RAM. 

So, for a great majority of the time it’s best to leave apps open. Even though they’re open, they’re really just sitting in a type of suspended mode that isn’t using any battery power whatsoever. Yes, the app is still loaded in RAM and taking space, but it’s not actually doing anything - it’s just lying there dormant. And, because it’s not using any battery power in this state, there’s no advantage to quitting it from a battery conservation point of view - especially if it’s an app you know you’re going to be using again and again throughout the day.

There’s also really no advantage to force quitting an app because iOS itself will do this for you if/when it needs to in order to free up RAM. So if you have 50 apps open and they stay open, then iOS hasn’t been pushed to the extent of needing to close any of them to free up memory.

Now, like anything, there are always exceptions to the rule (such as apps that have to perform background tasks). An example of this is one that plays music while you’re doing other things with your phone, or one that’s downloading content in the background, or one that’s counting how many steps you walk in a day, etc. However, iOS has an extremely efficient process for managing background apps/tasks and if you choose to quit these you’re basically saying you don’t trust the operating system to do its job properly.

So, feel free to quit your apps when you have to (e.g. because it’s frozen, etc), but don’t do it to conserve battery power. In fact, if you do, you’ll be achieving the opposite and using more battery power throughout the day!

Summary


You only need to quit apps if they're not working properly (e.g. an app has frozen, it isn't displaying properly, etc).
Apps listed in the App Switcher are not necessarily running - in fact most of them are not running at all.
Most apps in the App Switcher will be in a suspended state - this means they're not: visible on screen, executing code, using the CPU or GPU, or using the battery. However, they are still resident in memory until they are purged (if necessary) by the system to free up memory:


  Suspended - The app is in the background but is not executing code. The system moves apps to this state automatically and does not notify them before doing so. While suspended, an app remains in memory but does not execute any code. When a low-memory condition occurs, the system may purge suspended apps without notice to make more space for the foreground app.


Source: See Table 2-3 within the first reference link at end of this answer. 
The only apps actually running on your iPhone at any given point in time are the active app (i.e. the one visible on screen) and any others working in the background. (Note: There are some temporary exceptions to this in the case of apps that still need time to complete executing code they already started while they were active - typically this is only in the order of seconds but could theoretically extend to over a minute.).
In terms of apps running in the background, you can control which apps are permitted to do so (if they're open) by going to Settings > General > Background App Refresh. (Note: Just because you see an app listed here doesn't mean it will run in the background, but disabling it here means it definitely won't!)
If your iPhone is locked, then the app that was active (i.e. it was visible on screen) when you locked your device is now inactive. However, unless you've only just locked your iPhone and it's still finishing executing code or it's running in the background (e.g. playing music, etc) then it's not using the CPU, GPU or battery.


References


For more info about the various states an app can be in, refer to Apple's App Programming Guide for iOS: Execution States for Apps.
For more info about apps running in the background, refer to Apple's App Programming Guide for iOS: Background Execution.