because of the LAW of conservation of angular momentum of universe which is 0 in total sum. (you can use some other conservation laws too)
By definition these states are systems where the second electron is in the opposite configuration to the first. You can use photon-electron but you have to polarise photons which is not a problem - vertical or horizontal polarization is used in radar systems antennas. then you measure electron in in that plane , the same plane you released ElectroMagnetic wave/photon. electron is soo small that it has besides particle a wave-like properties too ..... its duality principle means it can be considered almost equal to massless photon....
remember if one of wave-particles is measured to be up along a given direction, then the other will definitely be down without the need for measurement. that particle will exibit particular north-south (or south-north) magnetic field as it moves and you can detect it. A pair of electrons or other combined particles in this state are said to be entangled.
if its pair hasnt been influenced you wont detect magnetic field because its spin is a smudge, its bluured not resolved in wave mathematical equasions, its everywhere....
All we need to do to produce an entangled state is to bring two electrons in close enough proximity so that their magnetic fields interact and then leave them alone. Eventually, the electrons will be entangled, after maybe having emitted a photon. This is due to the fact that the electrons in opposite configuration is the lowest energy state of a two electron system. We can also think about as the first electron being the magnet that prepares the second (or vice versa).
ofcourse you can have more particles entangled. ( many think inflation didnt happen and earliest radiation of space is smooth because everything was entangled.)