The short answer to this question is no, you don’t need the internet to listen to digital radio.
Digital radio and internet radio use completely different technologies and so generally can be listened to on separate devices.
Digital radio is broadcast as a series of waves of energy that are picked up by a receiver. These transmissions are in the form of DAB in the UK, although the later and more efficient DAB+ technology is available in other countries and may become the standard here although there are no plans to do this just yet.
A DAB radio is needed for these transmissions, giving the benefit of easier access and, in theory at least, better sound quality. These radios will also generally have an FM tuner for analogue transmissions.
Listening to Internet Radio
Internet radio is broadcast over the internet and can be listened to using any device that has a suitable connection. Typically, a mobile phone or tablet computer with a radio app is used to access thousands of stations worldwide.
A good quality internet connection is necessary, generally through your home broadband, to ensure a satisfactory listening experience and listening can cost money unless you have an unlimited data allowance.
Many stations broadcast programmes in digital, analogue and internet formats so you can listen through a choice of devices. Additionally, although a digital radio isn’t necessary to listen to internet broadcasts, you can buy a dedicated internet radio which will provide DAB, FM and internet listening through a single device.