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It's the other way around.
We observe that sometimes when a conductor is moved within a magnetic field, it generates an EMF, and sometimes it doesn't.
Then more careful observation shows that when it moves parallel to field lines is when it doesn't generate a voltage, and when it cuts across field lines it does.
So then we say 'voltage when it cuts field lines' to summarise what we observe.
Then we can get quantitive, and find how much voltage gets generated for what field gets cut how fast.
Unfortunately, these summaries and equations don't actually tell us what's going on, or why.
If you get down into quantum mechanics, then you start getting explanations of what could be happening at the magnetic field level. Unfortunately, that doesn't explain why quantum mechanics. You've just shifted the goalposts.
If it helps, a better summary of what's going on would be a loop of wire surrounds a certain amount of magnetic field, and it generates a voltage when the amount in the loop changes, either by the loop changing size (cutting lines of field), or by the field changing strength (like in a transformer).
Magnetic lines of Force Field, B and Magnetic Flux Field, H are present when current flows thru a conductor.
When released instantly , the current resists a change by inducing a large electric field transient V=LdI/dt. While current may now rapidly decay while V rapidly increases momentarily usually causing an arc when switched from contacts according to an L/R time constant until the extinguishing voltage is reached by some gap.
So in summary , a Magnetic Field H created when energized proportional to current and may be time varying or DC until released then that creates a larger Electric Field pulse , E which lasts for a brief time. The total energy stored in the inductor is being released when this current is cut instantly from the conductors and is released in the air between contacts in the form of a spark. Semiconductor diodes can normally suppress this with diodes to the opposite supply rail or through some RC snubber.
Even at slow rates of movement, the Lorentzian transformation describes how the relativistic warping of vectors will lead to "retarded" potentials, aka delayed potentials, of what occurred some tiny fraction of time ago. Faraday and Maxwell used "lines of flux" to describe the relations of the 3_D vectors.