Why did the TARDIS take them to the Doctor's childhood and why didn't she want the Doctor to know where they were?

The answer lies in the very same episode you just quoted. The TARDIS telepathic interface works in such way that it explores the "plugged" person's entire timeline in order to find the wanted destination, like it was explained earlier:

(The Doctor places Clara's fingers into squidgy sections on the Tardis console.)

DOCTOR: Just hold on tight. If anything bites, let it.

CLARA: What is it?

DOCTOR: Tardis telepathic interface. You are now in mental contact with the Tardis, so don't think anything rude.

CLARA: Why not?

DOCTOR: It might end up on all of the screens. The Tardis is extrapolating your entire timeline, from the moment of your birth, to the moment of your death.

CLARA: Which I do not need a preview of.

DOCTOR: I'm turning off the safeguards and navigation, slaving the Tardis to you. Focus on the dream. Focus on the details. Picture them, feel them. The Tardis will track on your subconscious and extract the relevant information. It should be able to home in on the moment in your timeline when you first had that dream. And then, we'll see.

CLARA: What will we see?

DOCTOR: What's under your bed.

(He starts the Tardis flying.)

CLARA: Ooo!

DOCTOR: Okay, now don't get distracted. Remember, you are flying a time machine.

(Clara's mobile phone rings, and the image of Danny greeting her at the restaurant flashes into her mind. As the Doctor walks around the console to her, he passes a blackboard with lots of items chalked on it, beginning with Evolution Perfects. It is what he wrote at the top of the show.)

DOCTOR: No, no. Don't you dare. No, don't. Don't, don't. Just ignore it.

Since Danny called Clara to her cell phone, she started thinking about Danny and that's why they ended up in HIS timeline, when he had that same nightmare, instead of hers.

Then, right at the end when the TARDIS ended up in the barn in Gallifrey with the child Doctor, the answer seems simple: Clara was thinking about the Doctor when she set the TARDIS flying, and once again, it (she) took them to the night the Doctor had that same dream.

As for why Clara didn't want the Doctor to know where they ended up:

As the Doctor himself had told Clara earlier in the same episode:

CLARA: Isn't it bad if I meet myself?

DOCTOR: It is potentially catastrophic.

CLARA: So why did you bring me out here?

DOCTOR: I was still talking. I needed someone to nod. Probably best for you to wait in the Tardis.

So yeah, it's potentially catastrophic if the Doctor finds his younger self, apparently (Except for those times when he has done exactly that, in the multi-Doctor episodes).