Physically, this principle (sometimes referred as to EEP = Einstein's Equivalence Principle) postulates that there is no experiment done in a small confined space which can tell the difference between a uniform gravitational field and an equivalent acceleration.

Put in another way, according to this principle, all physic laws wih hold the same form in a free falling Local Inertial Frame (LIF) in presence of the gravitation and in an Inertial Frame in absence of gravitation.

"The vehicle between flat spacetime and curved spacetime is the Equivalence Principle: the laws of physics are the same in any Local Lorentz frame of curved spacetime as in a global Lorentz frame of flat spacetime" Gravitation (Misner, Thorne, Wheeler) § 8.5.

Mathematically, this important observation states that in presence of a gravitational field, small enough free falling frames will be inertial, and that in these Local Inertial Frame (LIF), where the metric gμν reduces to ημν, the laws of physics from Special Relativity will hold true.

Refer to  the article The Equivalence Principle for more details.

Refer to the below youtube tutorial to get a good overview of the mathematical expression of the principle, i.e. how a local inertial frame (black colour) can be obtained by a general coordinate transformation at any point P of a manifold (blue colour)


Refer to the article 1907 Equivalence Principle first published expression to read the Einstein's first formulation of this principle.