Transaction ID Barcode

I am told that the barcode shows up automatically on the emailed receipt. To hone the question more, then, what do we do with it? In other words, when the theatre-goer arrives with a hardcopy of the email, do we scan it with a particular app on a smartphone? How does the validation then work?

It’s in the Help File – Section 8.1

You just need a standard Barcode reader to read the barcode. I bought a barcode reader with a USB interface on EBay which works fine.

Section 8.1 does not address the mechanics. Is the idea that we have to have a laptop or computer with us at the physical box office so that we can plug in a USB barcode reader? Is there no way to do this with a smartphone that has a barcode reader app and have the information automatically sent via the mobile signal to — and here it gets fuzzy for me…? Where exactly does the validation take place? On the PayPal server? On a particular CPU? On our website’s server? However that works, can a barcode reader be pointed at the barcode in the email on someone’s smartphone or tablet, or do they have to bring in a paper copy?

Sections 14.4 to 14.6 touch on the mechanics of Validation, although to be fair not in fine detail.
As you say, the idea is to have (say) a laptop, connected to the internet, at your venue. Sale Validation can then be carried out using the StageShow Tools page, and entering the TransactionID using the keyboard. Alternatively, if the barcode scanner is used to input the TransactionID, this will automatically be entered into the correct entry on the form, and the validation carried out. The validation is actually carried out by the web server, but the laptop used can be identified by the “Terminal Location” setting, which is specific to a particular terminal device (i.e. a particular laptop), and this will be recorded with the validation attempt.
The offline validator does a similar function (with certain limitations) but without the need for an internet connection.
The only involvement of PayPal is the allocation of the TransactionID.
To use a smart phone to scan the barcode would need an app that can read a barcode and pass it to a form on a webpage. Possible, but I don’t know anything that does it (somebody else may!).
My scanner didn’t like the barcode on my smart phone, but I believe this is possible.

Barcode shows up automatically on the emailed receipt, that means there is a built-in barcode generator? I haven’t seen or heard anything like this but it seems pretty cool.
After using barcode reader to do the reading part, then what??

Yes there is a built in Barcode generator!

This can be used to validate sales … details are in the Help file