Customize Apps Gallery in Appstore Page

How do some iOS appstore pages have customized backgrounds and styling?

I have the vague recollection that things might be changing in iTunesConnect to coincide for iOS 9, but for now the key to get the customized backgrounds & styling is indeed having your app featured.

And if that magic moment in your app's life does happen, here is a blog entry that instructs you how to handle the artwork requests.

EDIT: and actually, a few moments after I researched the answer to your question, I found a related (or totally duplicate) version of this same question that's been answered here before.

Custom image assets for use in the App Store

This is special functionality that is only available to apps Apple wants to highlight. If/when your app qualifies, they will contact you regarding asset submission.

For some apps there is a icon shown on top of Appstore Metadata, How can we edit it?

If you are talking about the "Essentials" badge, these are hand-picked apps by the App Store team. There is no way for you as a developer or publisher to configure it in iTunes Connect.

Same goes for the header image and background color Timur mentioned in the comments.

How to customize Apple App Store Appearance?

Some apps are featured by Apple. And sometimes, Apple asks to the seller to provide promotional artworks in order to promote an app on the App Store.

If Apple wants to promote your app, you will receive a promo art request and you will upload it through iTunes Connect.

Can Custom App Store Listings be created?

You can create different app store pages by adding additional languages to your app on App Store Connect.

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Simply go to App Store Connect > My Apps > Choose the app > Click on the languages dropdown on the right and add the languages. Then click on the version below "IOS APP" and you can add your language specific information.

That's exactly the same why as you would do it on Google Play Dev Console. Custom listing are something completely different as they allow you to target more specific audiences. The general languages correspond with the user language / store front.

How to add an App Store product page banner

There isn't an option for this on App Store Connect. Apple reaches out to developers for whom they promote on the store or whose apps are featured in Stories on the App Store.

Read over the page here about being featured on the App Store: https://developer.apple.com/app-store/getting-featured/

Side note: you'll notice that most apps with App Store banners are either (A) from large companies (like Google, as noted in your question) or (B) have been repeatedly featured on the App Store (like 1Password).

How to add the Cover image of your iOS App in the App store

Only featured apps can have customized cover image or header. These applications are selected by Apple. This isn't a setting that we can control.

Here is an article on how to get featured in the App Store:
https://thenextweb.com/dd/2013/10/29/get-featured-app-store/