How is the existence of Lorentz force explained only by analyzing the generated field lines? I cannot figure out how Lorentz force occurs only by analyzing the following diagram. 

Geometrical analysis is preferred.
 A: This is quite unconventional, but it looks like it's using the stress tensor. You know electromagnetic fields carry energy, and they carry momentum as well. The flow of that momentum is described by the Maxwell stress tensor
$$\sigma_{ij} = \frac{1}{2 \mu_0} \left(2 B_i B_j - B^2 \delta_{ij} \right).$$
The diagonal element $\sigma_{ii}$ of the stress tensor may be interpreted as the negative of the pressure in the $i^{\text{th}}$ direction. In your case the field is only in the $xy$ plane, so $B_z = 0$ and 
$$P_z = - \sigma_{zz} = \frac{B^2}{2 \mu_0}.$$
Since $B^2$ is higher above the wire, there is a net pressure pushing the wire down, explaining the Lorentz force. A simpler, heuristic way of understanding this is to think of every magnetic field line as carrying a tension, which is how Faraday originally understood it. That means they want to straighten out, so they push the wire down.
This method is unconventional because usually the stress tensor is derived starting from the Lorentz force law, but in principle the logic can go this way too. 
