As bizarre as it may seem, space itself is expanding - specifically, the vast regions of space between galaxies. According to Einstein, space is not simply emptiness; it's a real, stretchable, flexible thing and not just a model. In fact, understanding the properties and behavior of space is a major goal of modern physics.
The notion that space is expanding is a prediction of Einstein's theory of gravity, which describes a simple but universal relationship between space, time, and matter. But it was a prediction that Einstein didn't believe; in fact, he tried to modify his theory to get rid of it.
In the late 1920's, the astronomer Edwin Hubble first observed that distant galaxies are moving away from us, just as would be expected if the space between galaxies were growing in volume - and just as predicted by Einstein's theory of gravity. Since then, astronomers have measured this recession for millions of galaxies. But there's other evidence as well, the galaxies sit more or less passively in the space around them. As the space between galaxies expands, it carries the galaxies further apart - like raisins in an expanding dough.
Cheers.
No Einstein never said space was a "thing" of any sort. Spacetime is a model of interactions. The metric expansion of space is intrinsic, and can only be defined by energy and matter, as we cannot directly observe space itself as anything other than a relation of objects.
And your impromptu elementary history lesson does nothing to rebut the facts.