One simple explanation is that there is software for various access points, whether they are a computer or a smart phone. The idea is to transfer digital information between two digital wallets.
After downloading the requisite phone application (there are several, such as Blockchain), users follow the app's instructions. Typically, a merchant has a QR code that can be scanned with a smart phone camera. This is his public address. The user sends Bitcoin, or BTC, using the app instructions. The user's digital wallet then will show fewer BTC and, voilà, the merchant's will show more.