How do contactless payment cards function in practice?

Contactless cards essentially use the radio-frequency to communicate with each other - yes, its wireless radio antenna communication in the simplest of terms.

Source: Inside NFC: secure payment technology

The reader essentially illuminates the smart-card with energy so as to power it on and then be able to communicate with the card.

A very thorough infographic explaining how NFC cards work explain the concept of contactless payments. Other solutions of contactless payments are variant of this basic methodology in one way or the other:

Source: 5 myths about contactless payments

There is a really nifty (short) course you can do that explains how this all works (its not a course per se), but more of a guided intro: Near Field Communications - Mobile Wallet

There is a lot of talk about how just by illuminating someone's wallet with a powerful enough radio antenna/output you could possibly read their cards. A British company called Peratech has made a nifty hack so that this is not possible, by applying a pressure point in the card to enable the circuit to be completed for the card to be illuminated and read:

Source: Award-winning British high technology firm has solution to growing problem of contactless card fraud

Here is a nice overview video about the Contactless payments explained:

https://www.youtube.com/watch?v=...

For further reading:

• Here is a great FAQ so to speak from the Smart Card Alliance on Contactless payments: Page on smartcardalliance.org
• Step-by-step Guide - Contactless
  
• MasterCard PayPass Overview
  
• MasterCard PayPass
  
• Page on paysmart.com.br