People who are not United States citizens very often have the burden of proof in many aspects of immigration law. This is true whether you are applying for admission to the United States at a consulate overseas or at a port of entry, or trying to obtain your green card or become a U.S. citizen. A non-citizen has the burden of proof to show they are eligible for a visa, admission or the immigration benefit they are seeking. Having the burden means that the non-citizen is responsible for proving the negative, i.e. that you are not inadmissible and not ineligible for the relief or benefit.
In the context of a non-citizen with a criminal history, having the burden means that you are responsible for obtaining your own arrest and conviction records and providing them to the government. The government can deny you admission, relief from removal or the benefit you are seeking if you are unwilling or unable to obtain the records and prove you are not ineligible for whatever you are seeking.
Normally, anyone can go to the courthouse where they were prosecuted or law enforcement agency that arrested them and request copies of the certified documents that the immigration officers will want. However, this is not the case if the non-citizen successfully had their criminal arrest expunged or criminal case sealed.