How prestigious is a Fulbright scholarship
Yes, it is very prestigious, yet in many countries most non-academics will not have heard of it.

Regarding the benefits, I assume you did your research before applying so I don't know what we can tell you that you don't already know.

However, since this could help other people thinking of applying to one of the Fulbright programs, I am providing an (unofficial and incomplete) list of good things about this program. Please contact your local Fulbright office for more detailed and updated information.


Once awarded, you'll attend a workshop/training in your home country, that will help prepare you for applying to US institutions and for studying and living in the US. This can be very important for people traveling to the US for the first time.
If you need help improving your English, they'll likely send you 1 to 6 months ahead of your MS studies, to attend an English program in a US University (not the same university in which you'll get your MS).
They have academic advisers that will help you apply to Universities, will check your recommendation letters and statement of purpose, so that you can maximize your chances of getting into a top program.
You'll forever be part of the "State Alumni" network and will be invited to attend networking opportunities with other State Alumni members in your home country (and I guess other countries if you relocate).
During your first year of studies, you'll be invited to a 3-4 day Fulbright Enrichment Seminar on "topics of global importance" (topic changes yearly). Personal note: for me, this was an amazing experience and probably the best part of the whole program. This video shows what the grantees do on these seminars.
You may get some additional financial help (in addition to tuition), like a stipend for books and/or to buy a computer.


EDIT (Added the following):

On the intangible benefits of the Fulbright Scholarship

Ultimately, the Fulbright program is an International Exchange Program. It is hard to try to explain the intangible benefits you get from participating in such a program, but these quotes from Senator Fulbright can help:


  "International educational exchange is the most significant current project designed to continue the process of humanizing mankind to the point, we would hope, that men can learn to live in peace--eventually even to cooperate in constructive activities rather than compete in a mindless contest of mutual destruction....We must try to expand the boundaries of human wisdom, empathy and perception, and there is no way of doing that except through education."
  
  "The essence of intercultural education is the acquisition of empathy--the ability to see the world as others see it, and to allow for the possibility that others may see something we have failed to see, or may see it more accurately. The simple purpose of the exchange program...is to erode the culturally rooted mistrust that sets nations against one another. The exchange program is not a panacea but an avenue of hope...."


Very often, international Master's students in the US end up interacting primarily with other international students in their own program, and sometimes even with mostly other students from the same home country. A student participating in the Fulbright program has numerous opportunities to interact with US citizens and other international students from different programs and from different universities. It is an enriching experience that most likely will contribute to give you a more broad view of the world. This will likely have a positive impact in your career, either directly (for people in law, international relations, government, etc.) or indirectly (like helping you become a better teammate, a better project leader or manager).

If you want some examples on how the grantees experience their time in the US, this book has good reviews (I cannot comment myself, since I have not read it).