The setup seems even stranger considering this group of astronauts came to investigate a moon base that was established 10 years ago to do a mineral survey... but its inhabitants are now presumed dead after the catastrophe (which also happened ten years ago). There must have been ships more modern than the space shuttle to take them to the moon in the first place, not to mention some way to send them supplies... but the plot skips past this question to bring us space spiders, which currently inhabit the cobweb-filled base.
Because the group doesn't realize they're in a horror movie, they split up, and the two who leave the ship are, of course, killed by space spiders, which attack in Alien facehugger style. The Doctor does finally show some emotion when he thinks Courtney is about to be killed... but she manages to save herself, spraying the alien attacking her with some disinfectant she had in the school backpack she's been carrying with her.
Surprise! This proves that these aren't space spiders after all, they're really big germs that are on something really big. The moon is actually an egg and the increase in mass is because it's growing and, now, getting ready to hatch. Lundvik thinks they should kill the creature before it hatches while Clara and Courtney are both against murdering an unknown alien. But the choice, no matter how much weight is put on it, seems an odd one: wouldn't detonating nuclear bombs on the moon destroy the moon in the same way letting the egg hatch would destroy the moon? Lundvik, suddenly an expert on giant alien eggs, says letting it hatch would release a potentially dangerous alien and send egg shell fragments hurtling towards earth which would wipe out humanity... but detonating all of the nuclear bombs, though you'd have one less alien to deal with, doesn't seem like it would cause less damage.
In another callous move, the Doctor decides that this is a decision about the human race that has to be made by humans -- "It's your moon. It's your choice." -- so he hops in the TARDIS and leaves... even though none of them have a way off the moon whether they detonate the bombs or decide to let the egg hatch.