How did the Time Lords put a whole "Star" in a Tardis?
Do they have some kind of galaxy shrinking technology or what do they use?
Time and Relative Dimensions In Space - TARDIS
The catch here is that's not just a star, it's a collapsing star. From Journey to the Centre of the TARDIS
DOCTOR: The Eye of Harmony. Exploding star in the act of becoming a black hole. Time Lord engineering. You rip the star from its orbit, suspend it in a permanent state of decay.
This type of star is called a Eye of Harmony. The Doctor's version is a mere copy, with the original residing on Gallifrey. Apparently Rassilon was capable of such a feat
According to the ancient records, the legendary Time Lord Rassilon captured and placed the Eye – the nucleus of a black hole – beneath the Time Lord's citadel on Gallifrey (the Panopticon). Rassilon placed the black hole's nucleus in an eternally dynamic equation against the mass of Gallifrey and so that as long as the Eye existed in this "balanced state", the power of the Time Lords would "neither flux nor wither". However, it remained there so long that it passed into legend.
As was Omega
Omega was a solar engineer who, in trying to provide an energy source for the Time Lords' time travel experiments, harnessed the power of a supernova.
From The Satan Pit
DOCTOR: I'm just towing you home. Gravity schmavity. My people practically invented black holes. Well, in fact, they did.
The implication is that these ancient Time Lords knew how to manipulate gravity and space in fantastic ways. As such, moving a star into something the size of a phone booth would have been trivial.