If Gibbs energy of activation is positive, how is the change to the transition state possible?
...according to the Second Law of Thermodynamics, entropy of the universe can only increase.


Well, there was a nice similar question in bio.SE about this. A system is not universe. The overall entropy of the universe increases, but that isn't true for only a particular thermodynamic system.

Also, no one said the transition state is stable!


  The transition state of a chemical reaction is a particular configuration along the reaction coordinate. It is defined as the state corresponding to the highest potential energy along this reaction coordinate. Wikipedia


(High potential energy = lesser stability)

This chemwiki.ucdavis page is also good to help reshape your understanding of second law of thermodynamics.