Originally I would think Lithuania would be rather against the idea of losing its property? Was this not the case or were they more attracted to the idea of another kingdom?
Since the end of the 15th century, Lithuania was engaged in a series of conflicts with the vibrant Muscovite Russian state that was in the process of "Collecting Russian Lands". Already 1/3 of Grand Duchy was lost to Muscovy at the turn of century and other losses (Smolensk, Polotsk) followed during the 16th century. In this conflicts, Lithuania often depended on Polish assistance (for example, Poland provided about half of the forces during the battle of Orsha in 1514) that was becoming more and more important as Russia was growing stronger.
Another factor was the opinion of the monarch; early Jagiellon rulers were not necessarily supporters of a closer union. In Lithuania they were hereditary rulers, while in Poland they were elected monarchs heavily limited by laws and customs of the land. By 1560s, however, it was becoming increasingly unlikely that Zygmunt August will father a successor , and his alliance with the middle nobility against the magnates meant supporting a program of reforms which included closer Polish-Lithuanian union. In 1564 he ceded his hereditary rights to Lithuania to Polish Crown.
During the proceedings of the Sejms of 1569 in Lublin (there were two separate one, Polish and Lithuania) there was still some opposition to the union, particularly among the Lithuanian magnates. However, the incorporation (by Polish sejm but with the support of local Lithuanian deputies) of Voivodships of Podlachia (March 15th), Volhynia (May 26th), Kiev and Braclav (June 6th) into the Crown of Poland left them with little choice: they could either accept the union or break it and be left with a weak, rump state with unfinished conflict with Russia. Thus the last opposition was broken.