The Apple Wallet app is just a convenient, central place to store passes and scannable items you might need. Just like your wallet, where you might keep your credit cards, airline frequent flyer program cards, health insurance cards, hotel loyalty program cards, and so on, the Wallet app is a single app in which you can store identifying items for quick and easy access and scanning.
While you won't be able to access all of your airline loyalty program information directly in the Wallet app, you can access boarding passes; while you wouldn't really use the Wallet app for full access to your hotel loyalty program account to book reservations and change account info, you could use it to quickly pull up the confirmation number of an upcoming stay; while you can't use it to check showtimes and buy movie tickets online, you can use it to store a ticket you've purchased to be quickly scanned once at the theatre . You might find it more convenient for quick access to things like boarding passes and such, rather than opening up the individual airline mobile applications.
So long as you load items in to the Wallet app beforehand, you don't need wifi, cellular, or any other data connection to access them. You do, however, need that data connection to place items in to the Wallet app in the first place. Depending on the way certain Wallet items are configured (and that's up to the company who "issues" you the scannable item, more than it is Apple), a data connection can enable certain features. For instance, I keep my Starbucks card enabled in the app, and when my iPhone's GPS detects I'm pulling up to my local Starbucks it pulls up my Wallet app Starbucks card so I can quickly scan it and pay for my coffee; as I pull in to the CVS parking lot, the Wallet app creates a shortcut to my Wallet app CVS card so it's ready to be scanned at the counter or drivethru for my prescription; when at the grocery store, the Wallet app uses GPS to know I'm there and pull up the grocery store card to get scanned at checkout. Certain other Wallet items might not need that data connection and access to GPS, and will instead by triggered by time/date - the airline knows when you're set to depart for your flight that day, so will make your boarding pass available on your device's lock screen ahead of time so a quick and easy click can open it up to be scanned (as opposed to unlocking the device yourself, opening the Wallet app, selecting the boarding pass, and getting it scanned).
The Wallet app doesn't exist for the iPad - it's just available on iPhones and iPods. The thinking here is those smaller handheld devices will be the ones you'll want to be able to quickly access things on, and sized best to use with scanners at airport gates or when approaching the TSA screening lines and such.
So really the best way to explain the Wallet app is to compare it to your actual Wallet - a single location in which you store limited, but readily-accessible items. In your actual wallet, you might keep credit cards you most frequently use in pockets and card slots that are easiest to access; the Wallet app will present items you might need right away in a quick and easy access way. You can't keep your entire health insurance explanation of benefits package in your actual wallet; you can keep your health insurance ID card in it, just as that ID card alone can fit in your Apple Wallet. It's not quite a replacement for the individual mobile apps for airlines, hotels, retailers, etc, but a single location in which to store useful items from those individual business and service providers for quick access.