You are a stateless person or a refugee with a Refugee Travel Document I-571 issued by the United States and wish to know if you must obtain a visa to travel in the Schengen area on your Refugee Travel Document.
Generally this varies both by the country you wish to travel to, and the country that issued the refugee travel document.
According to this official document (Microsoft Excel required), among Schengen countries, US refugee travel documents are accepted for visa-free short term travel to Belgium, Bulgaria, Croatia, Germany, Hungary, Iceland, Italy1, Slovakia and Slovenia. (Netherlands is missing from the spreadsheet entirely due to an apparent clerical error. Its visa exemption is apparently still active and listed in Timatic.)
However, if you enter Schengen via Germany and happen to take a short side trip to Italy this might not be a practical problem, provided you don't cross a passport control. While there are mostly no border controls within the Schengen area, there may be occasional random checks especially on public transport, and border guards in countries other than Germany may be unfamiliar with the situation. This case is so unusual that it's not really clear what would happen, though I would guess they would see that Germany admitted you and send you on your way. However, it is possible that you could be detained and removed from such a country as Italy. Note that there are still land border controls around Switzerland, primarily for customs purposes (while Switzerland is part of Schengen it is not part of the EEA).
And while you didn't ask, I expect someone will: The UK is not part of Schengen, and its rules are: The UK requires visas of refugee travel document holders, except those issued by the UK, and so you will need a visa to visit the UK.
1 Italy accepts the US refugee travel document, but whether a visa is required depends on the nationality shown in the document.