Unfortunately it seems QGIS (via GDAL) offers limited support for HDF-EOS5 (or HDF5 in general), and there are discussions about the issue in this GIS-SE. You may want to try these suggestions.
Let me suggest a workaround using a software Panoply to get the data out of HDF5 and convert it into CSV. (So that you can read it in QGIS as a delimited text data).
- Install Panoply and start it.
- Open the dataset and select your preferred variable and plot it. (Follow steps 1, 2, 3 below).

- You will see a nice plot, but ignore it for the moment and select Array tab.

- Select and highlight all data, then copy them (e.g.
Ctrl+C
in Windows), then paste the data to a text editor (e.g. Notepad++). Now save the data as a text file.

- Start QGIS and read the text file by Add Delimited Text Layer.
