That looks like Gimp 3 to me - is it ? This next is much the same in Gimp 2.10
Some of your concepts are wrong.
Transparency comes from an alpha-channel. You add one from Layer -> Transparency -> Add Alpha Channel. If that is greyed out then the alpha channel already exists.
Removing a plain background can be done is several ways. For your image:
Colours -> Colour-to-Alpha The colour to remove is set as white, no need to change anything. If another colour then there is a colour picker to use. The colours in the image are light and they get a bit removed as well, become semi-transparent. Use the Opacity Threshold slider to restore. In Gimp 3 that is a layer effect (fx) so tick the merge option.
60 second example:
https://i.imgur.com/qg2DP7J.mp4
Saving / exporting. Save your work as a Gimp .xcf which keeps layers, transparency ... Exporting - Not all formats support transparency. Use .png or maybe tiff. A jpeg does not support transparency and replaces that with the current background colour.