I would like to transform this data: https://opendata.dwd.de/climate_environment/CDC/grids_germany/halfyear/precipitation/ (it is Gauss Krüger Zone 3) into WGS 84 / UTM 33N in QGIS.

How do I do it? Just setting to UTM didn't seem to work.

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    Try to reproject your data/layer from EPSG:31467 into EPSG:25832. Check this 7.1. Lesson: Reprojecting and Transforming Data. – Taras May 8 '20 at 6:37
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    Setting is just telling QGIS "project the data using this CRS". Reprojecting is telling QGIS "convert the coordinates of this data from CRS A to B". – Erik May 8 '20 at 7:54