In recent history, some communist or authoritarian governments have come to power with names falsely proclaiming democracy. I had in mind the Democratic People's Republic of Korea, the Democratic Republic of the Congo, and the German Democratic Republic. Why do these countries feel the need to express how democratic they are with their names?
hi! you may be interested in this post on the DPRK featuring /u/koliano
Legitimacy. Legitimacy is the right to rule. A government relies on legitimacy in order to exercise its power and to maintain its monopoly on the use of coercive force.
You'll notice how a lot of monarchs use the term "by the grace of God" - these monarchs derive their legitimacy by the approval of the Divine.
A republic, like the US, or England, or Germany derives its legitimacy from the consent of the governed, or, if you like, from the electoral process. The leader of the US, England, or Germany rules by the consent of the governed, because he or she was chosen by the people.
So a government like the DRC, GDR, DPRK, claims that its government is legitimate because they derive their power from the will of the people. North Korea is interesting because they claim that the the electoral process does not represent the people, and that only certain chosen ones can legitimately rule of behalf of the people.
To answer your question - these countries claim that they possess the right to rule because of the consent of the people.