Question

What causes the electric field lines to curve at the ends of the two lines (a positive and negative charge lines)??

Answer 1

The lines are always curves, it is just more apparent (the curvature is greater, or radius of curvature is smaller) closer to the charged ends.

Explanation:

We often make the assumption that field lines are straight (e.g. between parallel plates) but this is a simplification to make calculation easier. Without that you would need differential equations instead of linear ones.

p.s. Hope I've understood the question correctly, it is quite general and hard to visualise the scenario.