At the moment 62 third-world countries are permitted to enter Germany visa-free and remain there for a period of 90 days within 6 months. During this period, visitors are not permitted to work, but they can engage in business.
I am trying to find out at the moment if you would need a working visa for a one month program like this. Best is if you check out the embassy of the country you are planning to do your fab exchange at and find out the visa requirements by explaining the program.
If you need an invitation letter, send an email to the fab lab exchange contact of the local fab lab.
Invitation Letter Fab lab erfindergarden
Blank Inviation Letter