You chose this link, because you are not a citizen of a Schengen region country and your country does not have a visa waiver agreement with the EU (but do double-check the full country list for visa waiver agreements)

2. Obtaining your visa: you need to apply for a long stay, or so called "D type visa" at a Hungarian Consulate. Your D visa serves as a pre approved residency permit. The following are links to download the application form and the the visa application procedure.

WARNING: Your application needs in fact be approved by the Hungarian Immigration Office in Budapest and sent back to the US or country of application, thus the entire procedure might be lengthy, especially if you apply for your visa outside the US. You are urged to send the Director of the BSM Program an email with details (when and where have you submitted/you plan to submit the application, etc. Please, indicate your gender.) Afterwards keep us updated about the status of your application.

The paperwork.

The above link and the letter you have received from the American office describes the documents necessary for the submission of your visa application. The Hungarian consulates will require them and you should also save and bring a copy of each document to Hungary, since the local Immigration office may require these as well.

You will need:
Make sure that the insurance policy carries your name or you have an additional letter from the provider that the given policy covers you as well.

The bank statement proving availability of financial resources should not be older than a month (you might need to get a fresh statement sent after you to Hungary).

There is an application fee that is more or less the same, whether you apply for your VISA in the US or elsewhere. The most recent fee is about $83, but before paying or sending a money order you should always double check.

Where to apply in the US?

You have to apply for a visa at the Hungarian Consulate responsible for your state.
This link shows Consular Representations by states of residency in the US
There are three consulates: one in Los Angeles, one in Washington DC and one in New York City. The links (all) contain their contact data and the states of the consular district they serve. BE SURE you apply to the one in charge of your state in the US (but we prefer NY in case you have an option, such as in case you have a home and a study residence, since the NY consulate is the most flexible and experienced with BSM).

Where to apply outside the US?

The following links contain all embassies and representations of Hungary.

3. After your arrival in Hungary: if you come with the D-type visa/pre approved residence permit, you should visit the Immigration Office to collect your residence permit and register your local address. You are NOT SUPPOSED to leave Hungary before the registration, since your visa entitles you for one single entry and for the next entries (into the whole Schengen region) you need the residence permit (that is, if you leave without the registration, you will have to apply again for a visa, which will cost money and might take 30 days).

Once you have your residence permit, you might travel with that freely within the Schengen region, independently whether you are an EU citizen, US or Canadian citizen or are required to get a visa to enter Hungary. BE AWARE, though, that some of the neighboring countries that are rather popular with past BSM students, such as Croatia, Serbia, Romania, UK, Russia or Turkey, are NOT in the Schengen region, and though you may freely return from those countries, you MAY NEED A VISA TO ENTER those countries. Check with their consulates in Budapest.