particles in our universe behave quantum mechanically. you can measure this explicitly. it tells us that certain "observables" of particles such as position and momentum can't be measured separately from each other, you can think of it as a vector, if you know more of the position you know less of the momentum and vice versa.

if you study the math of this, it forms something called a "Lie algebra", the generators of this algebra for certain types of observables are the pauli matrices:

https://en.wikipedia.org/wiki/Pauli_matrices

You can use these for the basis vectors in quantum computation. Basically you're using the nuts and bolts of reality to execute computations. These are called qubits.

The way you do this is to perform a sequence of unitary gate operations in this basis, which you can think of as transformations of the particles amplitude (complex valued probability-like waves). Once you performed a bunch of operations, you can measure the result (born rule) to get a non-complex value out. This is equavalent to wavefunction collapse in physics. In the double slit experiment its like trying to observe one path of a particle vs the other.

Anyways, if you construct your algorithm so that correct answers constructively interfere with each other, and incorrect answers destructively interfere, you can create a measurement to get answers that classical computers cannot do. This is because the amount of states in quantum amplitudes can be much higher than the number of states in classical computers, or even particles in the universe. So you can get speedups over classical computations in ways that people are still wrapping their noggin around.