I'm sorry, but you heard a rumor.
Unfortunately, U.S. embassies and consulates cannot process requests for this form of protection because, under U.S. law, asylum seekers can apply only if they are physically present in the United States (or at least at a U.S. border or other point of entry).
There is a common misconception that U.S. embassies and consulates are basically the same as U.S. soil. It is true that international law protects national embassies and consulates from being destroyed, entered, or searched (without permission) by the government of the country where they are located (the host country). However, this does not give those embassies or consulates the full status of being part of their home nation’s territory. Therefore, U.S. law does not consider asylum seekers at U.S. embassies and consulates to be “physically present in the United States” (or at a U.S. border or point of entry).
Although it is a plenary power of the President to state where asylum seekers could go for processing, and applying at the US Consulate or Embassy in Mexico for example has been discussed, mostly on the right, as an alternative to illegal immigrants being coached to ask for asylum when they get caught crossing the border.
One can apply for Affirmative Asylum Processing through the mail, in effect, requesting a review before an immigration judge, or at a port of entry.
There might be another related process that does allow the requester to do so in the host country of the US embassy or Consulate, the US refugee admission program. Same source as above:
How to Obtain a Referral to the U.S. Refugees Admissions Program from a U.S. Embassy or Consulate
You might be eligible for an embassy or consulate referral to the U.S. Refugees Admissions Program, which is basically a request by the embassy or consulate that another U.S. government agency (the U.S. Department of Homeland Security or DHS) examine you to decide whether you should be allowed to enter the United States as a refugee (a form of long-term protection very similar to asylum status).
This option may be available to high-profile figures or other people personally known by U.S. diplomats. However, even if you meet this condition, you might still need a personal referral not just from any diplomat but from the ambassador him- or herself, simply because you are still in your home country.