To leave and enter the US as a permanent resident, all you need is your permanent resident card. However, to enter most other countries, you need a passport. (I think only Canada allows US permanent residents entry with just a green card, and then only if traveling by land or sea. There might be others I don't know about.)
If you are unable to get a passport from your country of citizenship, as is the case with most asylees, you can apply for a refugee travel document from the US. More information about the application is available at https://www.uscis.gov/green-card/green-card-processes-and-procedures/travel-documents. A relevant excerpt:
A refugee travel document is issued to a person who has been granted refugee or asylum status, or to a permanent resident who obtained a green card because they were a refugee or asylee.
More information about the foreign travel can be found at https://www.uscis.gov/sites/default/files/USCIS/Archive/Archive%20News/2006%20Archive%20News/Asylee_travel_information.pdf.
You should be careful about returning to your country of citizenship, or applying for a passport from that country, since that could cause you to lose your asylum status.
A relevant excerpt:
... an asylee or a lawful permanent resident who obtained such status based on a grant of asylum status may be questioned about why he or she was able to return to the country of claimed persecution and, in some circumstances, may be subject to proceedings to terminate asylum status.