From the link:
"The irradiation process sterilizes mail by passing it through a high-energy beam--an electron beam or x-ray. This beam is ionizing radiation that delivers in a dose approximately 2 million times more potent than a chest x-ray."
The x-rays used to check parcels etc by Customs will also be more powerful than those for a medical x-ray, but can still leave electronic circuits undamaged, e.g. a laptop is scanned for drug concealment without suffering harm. However the sterilizing beam used to kill any biological threat like Anthrax in mail is powerful enough to damage the chemical bonds and even some of the atomic particle bonds that make things like flash RAM work, destroying it, or degrading it enough to cause premature failures.
"Radiation" is just transmitted energy. Our consideration is the type of damage it can do to matter it encounters, which is governed by how much energy the radiation possesses. Radiation from visible light to longer wavelengths such as infra-red is not considered to do damage, and is called non-ionizing radiation. Radiation beginning from mid Ultraviolet has enough energy to start causing changes to chemical bonds, the very shortest wavelength UV begins to be able to break atomic bonds, thus doing things like knocking an electron out of a atom producing an ion. This is called ionizing radiation, which continues up through x-rays and gamma rays with increasing energy.
With all of these described so far, a damaging source of energy only exists whist it is being supplied from an external source. Once this stops, the damage (i.e. to organisms like Anthrax) progresses no further. That's where the assurance on the website link comes in: "Irradiation does not make mail radioactive"
If an extremely high energy source is supplied, such as extremely energetic Gamma rays, some materials can undergo nuclear transmutation and acquire induced radioactivity. This is what we may hear of as nuclear radiation or radioactivity in objects or substances, such as radioactive Iodine in food. Even after the external source of energy is removed, atoms that were changed continue to emit (radiate) particles with enough energy to change nearby atoms, continuing to damage more chemical or atomic particle bonds for some time.
However, as long as the original irradiation is only as strong as it needs to be to kill biological threats, without taking the energy high enough to cause nuclear transmutation, nothing becomes radioactive.