A reasonably scientific answer {below} is offered by Gaelen Armstrong on Quora {at: https://www.quora.com/Reality-is-merely-an-illusion-albeit-a-very-persistent-one-What-did-Albert-Einstein-mean-here }. I am interested in hearing other answers (hopefully more from physicists than philosophers ;)

"Einstein is referring to existence as understood in quantum mechanics (/field theory/unified field theory). If you want to understand what that means it'll be useful to have a quick recap of the difference between Newtonian and quantum mechanics (so far as I understand it):

In classical physics, the world is perceived as a clearly defined realm of distinct and precise values. Matter is matter, forces are forces, and the physical world interacts like billiard balls – each unit being separate from the other. From this outlook, an object is one thing and not another. A particle is always a particle, and opposites are not connected. This is the world viewed on the gross, surface level - the world of classical physics (also called Newtonian physics or Newtonian mechanics, after good ol' Isaac).

With the advent of quantum mechanics it became apparent that the laws of classical physics do not explain behaviour on a quantum level. The movement of electrons, for example, can’t be explained by Newton’s laws of motion. Billiard ball mechanics no longer apply. Moreover, an electron appears to behave both as a particle and a wave. This phenomenon is called is a particle-wave duality.

At the quantum level, it is no longer correct to consider things from the classical perspective, as objects become increasingly more unlocalized and abstract. Fritjof Capra gave voice to this principle,

  • [The] distinction between particles and the space surrounding them loses its original sharpness and the void is recognized as a dynamic quantity of paramount importance.

From the quantum perspective the world is composed of mathematical concepts and approximations, not concrete realities. Subsequently {consequently??}, two or more characteristics can be present within one consideration. Electrons can be both a particle and a wave simultaneously. It is regarding this apparent sphere of contradictions that Robert Oppenheimer wrote

  • If we ask, for instance, whether the position of the electron remains the same, we must say ‘no’; if we ask whether the electron’s position changes with time, we must say ‘no’; if we ask whether the electron is at rest, we must say ‘no’; if we ask whether it is in motion, we must say ‘no’.

In quantum physics, the appearance and behaviour of a system changes as one goes from the world of particles, to particle-wave dualities, to fields (quantum field theory), and on to finer strata of creation (unified field theory). And, according to the correpsondence principle, as we go from classical physics to quantum mechanics to field theory to unified field thoery we are going to more "real" areas - that is, the reality of classical physics is quantum mechanical approximations. The reality of quantum mechanics is quantum field fluctuations. The reality of quantum fields is the the superposition of all possible field states of the unified field. 

Which brings us back to Einstein's quote. By "illusion", Einstein means it has the appearance of being real. By "persistent" he means that it is a continued appearance which is convincing. After all, we may all be quantum field fluctuations, but see how far that gets you when you walk into a brick wall. We think the concrete world around us is real, but it's really only the outcome of levels of physics which are "more real" - the world as we see experience it is not the world as it fundamentally is. It's an illusion, insofar as there are "more true" levels of understanding of physical phenomena."

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