There is a quantum electromagnetic field that permeates all of spacetime. When you energize a magnetic field, perhaps by switching on an electromagnet or perhaps by putting a permanent magnet into a place in space, you add magnetic flux into this field. Magnetic flux is sorta† like an electric current, but it flows in the field itself, rather than just through a wire. Electrons moving in the field feel a force from their interaction with the resulting magnetic field. If the electrons are moving through a wire, the wire feels the force. If the electrons are in a permanent magnet, the magnet feels the force. That’s the force that pushes two magnets together/apart.
† This is a rather extreme simplification. Electromagnetism is best described in equations, but that’s the antithesis of ELI5.
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