The only reason to define things as they are in classical mechanics is that they give rise to the correct equations of motions that can be directly measured and observed. Given a field theory described by $\phi(x)$, its action is defined as
$$
S(\phi,\dot{\phi})=\int_{\mathcal{D}}d^4x\,\mathcal{L}(\phi(x),\dot{\phi}(x),x)
$$
and the equations of motions for the fields are $\delta S=0$. This is an experimental result that can be observed at any level: classical mechanics, optics, electromagnetism, classical field theory, quantum mechanics, string theory and so on and so forth. In the case of a point particle in classical mechanics, the function
$$
\int d^3x\,\mathcal{L}(\phi(x),\dot{\phi}(x)) = L(\phi(t),\dot{\phi}(t),t)
$$
(referred to as the Lagrangian) turns out to be $L=T-V$ because in this way it generates the correct Newton's equations of motion, no other reason than that.
This equation seems weird, wouldn't E_k + E_p make more sense?
Why would that make more sense? That is the total energy of the particle, which is a quite different thing (although it can be related).
Is it some sort of work done? I sort of get that it works and derives newtons equations(?), but I cannot connect it to the real world
There is plenty of literature deriving the Lagrange equations starting from the Newton's laws, passing through the principle of virtual works taking into account constraints and degrees of freedom and so on. Usually the course in analytical mechanics covers extensively such topics; however, the action principle does come from the real world indeed, making use of a little formalism and generalisation.
