Allianz is a multi-national insurance company. The page you present solicits enrollment in travel insurance from Allianz in Italy. There's a bit more to it than just filling out the "card" on the first page.
When one fills out the card, one is taken to another page, where one must choose various kinds of coverage, or decline those coverages. Finally, the customer must actually pay for the policy. Standard practice would be to do so using a credit card.
After the customer fills out all the requested fields on the various web pages, and makes the credit card payment, the insurance will be in force for the calendar term specified and the coverage(s) selected. Allianz may well not even send you an enrollment card, but only provide the possibility of printing out a confirmation page.
Finally, Allianz is far from alone in this field: many, many companies sell travel-related insurance of many types. Make sure you read all the pages to understand what might be covered, and what might be excluded.
Many buy insurance; many travel without it. Whether this is "worry-free" is a matter of opinion for you alone.