The exploration cost bonus decreases the amount of Tar that is necessary to fully explore a Crypt. Crypt exploration effectiveness, on the other hand, increases the exploration progress that each march makes. The chance to double-explore a Crypt grants a probability that a march will explore a Crypt twice without spending any extra Tar.
The base maximum march time to a Crypt is limited to 20 minutes. Various technologies and Hero talents allow you to increase this time, however. Also, please note that the Ayrin avatar removes all such restrictions immediately.
In order to explore a Crypt faster, you can take more Tar on a march than the base amount required to explore the relevant Crypt. This also grants you a 10% bonus to exploration efficiency. Please note that in this case, the game first divides the amount of Tar taken by the base amount required for exploring the Crypt. It then adds the 10% bonus to every "strike" performed. Only after that does the game apply the exploration efficiency bonuses granted by technologies, talents, titles, etc.!