I'm gonna give it a shot and give an analogy with a comp. science audience in mind.
All Quantum Field Theories are effective theories. Effective means that they work up to certain energy-range, they do not intend to be fundamental.
For example, the Fermi theory of beta decay is an effective theory that works only up to the energy of W and Z bosons. Quantum Electrodynamics (QED), the theory of electromagnetism and photons, is an effective theory which is valid up to the electroweak scale (~250 GeV). So on and so forth. All of them are effective, and therefore break at some point at higher energies or, equivalently, shorter distances.
Renormalizable theories are such that we can abstract away the physics beyond that breaking point where the theory doesn't make sense anymore. And capture the physics beyond that point into a redefinition of a few fundamental constants that we can take from the experiment. To rehash the basic idea, the theory doesn't work beyond certain energies. In principle physics beyond those energies impacts our predictions, because in QM you must account for all the processes. But renormalizable theories are nice enough that we can put physics beyond that energy scale behind a black box and we just have to redefine a few fundamental constants like the mass of the electron.
Comp. science audience analogy: It's as if renormalizable theories gave us a neat API that does not leak the ugly internals of what happens at super high energies (short distances). Like you don't need to know machine code or assembly to import Pytorch and build a neural network in a few lines of code.
For gravity this doesn't work. Gravity is mediated by a massless spin-2 field (Einstein's theory). If you try to quantize this theory you will find that it explodes at second order. We can calculate the first quantum corrections to Einstein's gravity but then it explodes. And when we attempt our tricks to hide all the short distance/high energy stuff inside a black box (renormalization) it just doesn't work. It's as if the API was leaky. It leaks the internals to the high level.
To give you a couple of examples of this leakiness. Dark energy, the energy of the vacuum, causes an expansion to our universe at the largest scales possible. So it's a phenomena of super loooong distances and yet, it's dominated by the super small distance (high energy) interactions that contribute to this energy of the vacuum. Another example, black holes are typically hyper massive, huge beasts and yet... they are inherently quantum gravitational objects for which we need the full theory of quantum gravity to understand them.
This is a blessing and a curse. The curse is that it makes our jobs of getting a theory of quantum gravity so much harder. The blessing is that it gives us a chance of peeking at a more fundamental theory of physics. If we could have worked out everything with effective theories we may never know what's beyond those black boxes that hide the internals. Gravity gives us the chance to peek through and understand something deeper.