Yes, it's true (but imprecise). All residents of France (this is unrelated to citizenship) need to carry an authorization and the police is checking these authorizations. The authorization is a sworn statement (“déclaration sur l'honneur”), i.e. a self-statement, or a declaration by the employer. What's not clear in this short statement from the BBC is that this is not government-issued paperwork: you or your employer has to fill the form, you don't need any kind of government stamp.
The applicable law is décret n° 2020-260 du 16 mars 2020. I'll translate the main disposition:
In order to prevent the propagation of the covid-19 virus, until 31 March 2020, no person may travel outside their domicile, excepting for the following purposes while respecting general measures to prevent the propagation of the virus and avoiding any grouping of persons:
- Trips between one's domicile and a place where one carries out professional activity and professional trips that cannot be deferred.
- Trips to buy supplies that are necessary for professional activity and to buy basic necessities in establishments whose activity remains authorized (…).
- Trips for health reasons.
- Trips for essential family reasons, to assist vulnerable persons or for child care.
- Brief trips close to the domicile related to individual physical activity, excluding any collective sports, and for the needs of pets.
Persons who wish to benefit from these exemptions must, during their trips outside their domicile, carry a document justifying that the trip is within the purview of one of these exceptions.
Décret n° 2020-264 du 17 mars 2020 sets a fine. An “amende forfaitaire” (article 2) for “contraventions de la 4e classe” is “135 euros maximum”. Note that regardless of the fine, being out without a valid purpose is illegal and the police may force you to go back home (so it's not a €135 day pass).
There's a more detailed official information page (2020-03-19 08:30 version), prominently linked from the Interior ministry home page (archive link). Only a very short summary is currently available in English on the French government site.
As of March 17th at noon, outings will only be authorized, with a certificate, to:
- Go to work, if remote working is not possible;
- Go to your local grocery store;
- Go to a medical appointment;
- Take your children to daycare or to take care of an elderly person;
- To work out close to your home.
Download your personal certificate here
Download the employer's certifcate here
As indicated in the official information page linked above, the personal certificate must be printed out or written out on normal paper, and you must fill a separate one for each trip. The employer certificate is valid for the whole duration.