Here is a detailed summary of the Italian article on Vanity Fair (I hope a summary is allowed by copyright rules):
Andrea Casiraghi met Mathieu Chatelin by chance, during a political science lesson at the American University of Paris; Mathieu says they were seated next to each other and Andrea helped him taking notes. They eventually became very good friends (he says Andrea often goes to visit him in Paris and they talk about soccer and politics and cook together) and Andrea got very involved with Mathieu's illness and the Motrice foundation.
The interviewer met Andrea before the dinner at Chez Maxim, dragged him to take the pictures that go with the article and then asked him a few questions.
In the interview ANdrea says he got interested in the issue mainly through his friendship with Mathieu, but as time went by and he learnt more and more about this disease, he got concerned also as a person and a potential future father (not that Tatiana is pregnant). He says he wants to became, one day, a good father, and that he is grateful to Mathieu who made him see a new side to life and to suffering; he takes Mathieu as an example on how to live and be active and loved despite life obstacles.
The interviewer asked him if he takes after his father in his spontaneous way of connecting with others; Andrea replied the question was too difficult to answer (maybe because he was so little when Stefano died) but that he learnt the value of social and charity efforts from his mother's example.
He said he was proud his whole family was at the dinner to support him (during his speech at the dinner he also particularly thanked Tatiana for her support), and that he wants to raise money through happy events because sick people need joy, not sadness and an "air of tragedy", but that he takes the foundation and its scientific efforts very seriously.
Andrea said using his popularity to help a cause he believes in is the only thing he can offer and that despite being very shy, and suffering from being always scrutinized and in the spotlight, he doesn't have a right to complain, when compared to the situation of many ill people who suffer every day. He said what he finds very difficult is not being allowed (by the public) to make any mistake; but he is learning to manage his situation and is doing his best not to let people down and to live up to expectations.
The interviwer asked if after studying Political Science in Paris and at NYU he'll continue being involved in charities and Andrea said he will surely continue and that his journey will probably evolve in accordance with the Principality traditions; he says he is careful, takes life step by step but hopes to achieve good results. He ends saying he loves Italy, that he considers it his country, and that he regrets he can't live here at the moment.
There is also a very small interview to Tatiana who basically says she is very proud of Andrea, that neither of them like being in the spotlight (she stressed the point that Andrea is very shy) but that they both believe in what they're doing to raise money and awareness and that they both love Mathieu.