Physics, partly.
MRI is all about the protons. Under normal conditions, the protons in your body are all spinning ('precessing') around their own axis, but they're disorganized: each proton's axis is pointing in a different direction. They're like little wobbly tops drifting through space.
In an MRI scanner, the strong static magnetic field (B_0) forces the protons into alignment, so that their rotational axes are now lined up with the field's north-south axis. The field needs to be very strong for this to work, which is why MRI systems usually have expensive superconducting magnets.
Now that we've created a nice organized system, we're going to destroy it. A quick radio frequency burst energizes the protons and 'knocks them over' so they're no longer aligned with the field. Once the pulse ends, they 'relax' and realign themselves with the magnetic field, releasing some of that energy as they do so.
Sensitive detectors around the subjects' head detect those emissions and use it to determine how long it took for protons to realign themselves with the different components of the magnetic field. T1 is the time (or formally, the time constant) needed for relaxation parallel to the static field; T2 is the time needed for protons to relax to the transverse component. The T1 relaxation time for fluids is on the order of seconds, while fatty issue is more like 50-150ms. In the brain, grey matter has a relaxation time of 1.3 sec, but the fat-coated white matter relaxes much faster (~0.8), which makes T1 images very useful for examining brain anatomy.
Hopefully, this little crash course has revealed one of the bottlenecks in MRI: the actual signal being measured is slow.
Of course, I've massively oversimplified things and didn't explain at all how we localize these responses. The proton's precession frequency depends on the magnetic field, so by changing the static field slightly (across space), we can measure T1 at different locations, and sometimes even overlap measurements. You can't switch the field too fast though, or you'll start to induce currents in the subjects' nerves, which hurts. This is actually the principle behind a brain stimulation technique called transcranial magnetic stimulation.
There is still tons of room for improvement. Stronger fields lower the relaxation time, so the scans are faster (and the relaxation time estimates are better). Improvements in the RF coils help a lot too: the signal being measured is very faint and there's a lot of self-cancellation. On the software side of things, a lot of effort has already gone into designing clever pulse sequences—and the sophisticated signal processing needed to interpret their results.
Things will obviously continue to get better; I was just looking at some data from ten years ago and it looks awful compared to more recent scans.
That said, the "just use machine learning!!!" tone in some of the comments is kinda frustrating. Most of the people in this field aren't dummies--if it were as easy as downloading PyTorch, someone would have done it already. It turns out that the biology and physics are both stupendously complicated (and fascinating too).