The most important requirement to obtain asylum in the United States is to show fear of persecution in your country or fear for your life. This cannot be done inexactly or lightly, and the fear must be well established. Supporting documents help a lot, but it is true that you can win an asylum case based only on your testimony and witnesses to the persecution in your home country, in order to establish that you have suffered persecution in the past.
However, even if you have not suffered persecution, but you are able to demonstrate well-founded fear that you will be persecuted or that you fear for your life, you could be awarded an asylum visa. The great advantage of demonstrating past persecution is that it then becomes the responsibility of the United States Government to demonstrate that the circumstances in your home country have changed, if it wants to deny the case. If a past persecution cannot be demonstrated, the burden of proof lies with the beneficiary to properly document the reasons why he thinks that he/ she will suffer persecution in his/ her home country.
If you are in the United States legally, you can submit an asylum petition and remain in the legal situation, even if you are denied the case, until the legal situation you are in is over. But if you entered the country illegally or are in the country illegally when you submit the petition, you will be subjected to deportation proceedings if the application is denied. The point is that once you are subjected to deportation proceedings and go in front of a judge, you reapply for asylum, but also request cancellation of deportation and relief under the Convention Against Torture (Convention Against Torture) mechanism, so that you still have a chance to fight your assylum case.