Article 157ter of the LIR allows non-residents to be taxed as Luxembourg residents. It is not an administrative status to obtain: it is an option to request, subject to income conditions.
The stakes are concrete: without tax assimilation, most Luxembourg tax deductions remain unavailable to you. The conditions differ depending on whether you live in Belgium, France, or Germany.
Assimilation to Luxembourg resident status depends on your country of residence (rules differ between Belgium and France/Germany).