When is a LTI IIR $H(z)$ system minimum phase
but what does the stability mean when talking about a system?

It means that all poles are INSIDE the unit circle.

and if so how does it turn out that when a system is causal and stable its also min phase.

Sorry, you got this wrong. Causal, stable and LTI does NOT imply minimum phase. A simple counter examples is a one-sample delay. It's causal, stable and LTI but definitely not minimum phase.
Minimum phase requires that all zeros are also INSIDE the unit circle as well.
Update
When you invert a system the poles become the zeros and the zeros become the poles. A minimum phase has both all zeros and poles inside the unit circle, that means that poles and zeros of the inverse will also be inside the unit circle and hence it's also minimum phase.
A non-minimum phase system has zeros outside the unit circle. If you invert the system, this zero turns into a pole that's outside the unit circle and hence the inverse is unstable (or non-causal).