How mechanical Strain developed in metal bar at molecular level?

It is in my view a very deep question, and a detailed answer depends on specific material classes.

At a high level, some understanding could maybe be obtained by noting that generally bonds have to re-align themselves in the direction of applied load. At a macroscopic level this manifests itself as a resistance to volume changes, the one which in engineering elasticity is modelled via the bulk modulus: resistance to shape changes is instead modelled via the shear modulus.

As changes in volume cost energy, a stretched metal bar will attempt to reduce such penalty by decreasing the cross section. I give for granted that the reasons behind the stretching in the direction of load application are obvious.

In this frame, it is also interesting to note which materials will react to stretching by contracting (in the plane perpendicular to the applied load) to such extent, that the volume before and after the stretching is virtually the same. These are rubber , loosely speaking with Poisson Ratio $nu approx 0.5$. indeed, rubber elasticity relies mainly on macromolecular, long chain re-alignment, and not on bond stretching.

The page What is Poisson's Ratio could be interesting, the video of the stretching honeycomb is revealing.

Metals have a crystalline structure. Let's take one type of lattice: FCC.

That arrrangement is the most compact possible, if we model the atoms as spheres. Only the Pauli exclusion principle avoids they being closer than they are.

The effect of a macroscopic tensile strain is to increase the interatomic spacement in the stress direction, distorting the lattice.

The deformed lattice has now some space for a rearrangement, and the transverse contraction is the result of the neighboring atoms filling the blanks so to speak, until reaching the limit of the exclusion principle.

But, as the fcc lattice is the most compact possible arrangement, any other, as that distorted one, will have a bigger volume. The Poisson coefficient is then smaller than $0.5$ (what would result in a constant volume).