How would this work with the ideas of physic we already understand? I would like to see the mathematics written out and please don’t send me to an unfamiliar website to read.
Rchter65 has stated the fact with few words.
For quantum mechanical states, and particles in this sense are quantum mechanical entities,their behavior in space time is described by solutions of quantum mechanical equations (and by special relativity of course).
These solutions, called wave functions are functions of the fourvectors of the particles , i.e. energy and momentum and of the quantum numbers describing the particles, as is spin. Conservation of quantum numbers is mainly what the term "entanglement" carries in quantum states.
For example, take the $π^0-> γ +γ$ . The $π^0$ has spin 0 in its wavefunction. The wavefunction after the decay for the two $γ$ from conservation of angular momentum has to have the gammas emitted with opposite spins ( a $γ$ has spin + or -1 to its direction of motion). When you measure the spin of one $γ$ you immediately know the spin of the other,because they are entangled with their wavefunction describing the decay of the $π^0$ and because of conservation of angular momentum.
So it is all in the mathematics of wavefunctions and the conservation laws and quantum numbers.
Please note that this is not special to the mathematics of wavefunctions. The same logic can apply in every day situations. Take a pair of twins, John and Harry and you know that one of them works in London and the other in New York. If you meet for work with John in an office in London, you know immediately that the one in New York is Harry.
Entanglement simply means that you have two (or more) particles described by the same wave function.