is there a pattern to why a country's leaders would collaborate with the enemy instead of trying to free their people?

Leaders often place a very low priority on "freeing their people" - why else would dictators who use force to stay in power exist? Even in the absence of war with an enemy, or defeat by an enemy, there are people who place their desire to become a "leader" far above the benefit of the people they hope to "lead". Why then should collaboration with an enemy, despite disadvantaging their people, be surprising? Some collaborating leaders simply use the enemy as a means of climbing to power - they become a puppet dictator, rather than a dictator without an overlord, but the basic principle of personal power before benefit to the people is the same. You already named an example of such a person: Vidkun Quisling.

At the other end of the scale are leaders who collaborate with an enemy because they think it is the best way to protect their people. For example, Pétain and the French government of the time sought an armistice to avoid the destruction of much of the not-yet-invaded parts of France, and to avoid the complete occupation of France by the Germans (which might have been followed by either direct German rule, or rule by a German-chosen puppet government). This kind of choice - to surrender to an attacker rather than continue to fight - has been very common historically. For example, a city might submit to an invading Mongol army, and the population continues to live largely as before, rather than dying in a siege or storming or sacking of the city. The invader might leave a small garrison, and a governor to supervise the city administration (which would often continue to run as it had before the invasion). If taxes now go to the invader rather than the original king, how much difference does that make to the people in the city?

There can be disadvantages to such quick surrender - the original king might be quite offended by such a display of disloyalty and consequently destroy that city themselves. The same applies to vassals of King A who defect to King B (e.g., when King B invades) - King A might decide to reconquer those lost territories. There are many cases where leaders considering such surrender or defection misjudged either the short term or long term risks, but there are also many cases where they chose successfully, and protected their city or territory from much destruction. If there is no overlord, surrender is safer (the British did threaten Pétain and the French government (e.g., that Britain might need to bomb French ports if the Germans used them), so allies still need to be considered, even if there is no overlord).

In some cases, leaders have pushed their nations to strongly resist an attacking enemy, sometimes emphasising "freedom" in their rhetoric. Two notable cases of continued resistance against an enemy from World War II are Germany and Japan. Germany resisted rather than reaching an agreement with the Allies until most of their territory was over-run, and Berlin captured. The new leadership (after Hitler's suicide) quickly reached agreement with the Allies (in the form of accepting the demand for unconditional surrender). Japan resisted long after the war was evidently lost, despite the destruction of many cities by (conventional) bombing. With two atomic bombs and the Soviet entry into the war, they changed their minds and accepted the almost-unconditional surrender demanded by the Allies. A question: did this protracted resistance benefit the German or Japanese people, or would their people have been better off if their leaders had surrendered earlier?

If a government believes that they can continue to run their country (even if under a new overlord), and avoid widespread devastation of the country, and therefore collaborates with the enemy, this can be a quite rational choice. Of course, if one believes strongly in a "better dead than Red" principle, annihilation of the country might be seen as the lesser of the two evils.

Thus, collaboration with an enemy can result from a huge range of motivations, from a desire for power regardless of what is good for the people to surrender and collaboration from a sincere belief that it is in the best interests of the people.

In practice, the collaborators usually have mixed motives. Even at the benign end, the leadership might see collaboration as the only path that allows them to retain some power (even if protecting the country from devastation is their main motive). Collaborators can be motivated by monetary greed as well as greed for power.

For a look at some of the complexity of collaborating with an enemy, both at the level of leadership, and at smaller scales, see:

• Barrett, David P.; Shyu, Larry N., eds. (2001). Chinese Collaboration with Japan, 1932–1945: The Limits of Accommodation. Stanford University Press.

Another interesting case you might like is that of Puyi, the "Last Emperor", who had been the last Qing emperor of China as a child, and was made a puppet emperor of Manchukuo (i.e., Japanese Manchuria). Thanks to his autobiography, we have a view of collaboration from the inside:

• Henry Pu Yi (2010). Kramer, Paul (ed.). The Last Manchu: The Autobiography of Henry Pu Yi, Last Emperor of China. Skyhorse Publishing.