How is the neutron capture cross-section increased in thermal breeder reactors?

Thermal neutrons get lots of attempts. What you need, which the wiki article doesn't cover, is a picture of the overall "neutron budget" inside the reactor at a given instant. Some reactors run in the 98% neutron efficiency range. With fission averaging about 2.5 neutrons produced per fission event, there is plenty of headroom (at least on paper). You need good moderators and perhaps a hotter fuel load. Plus an external neutron accelerator to help manage both power output and breed ratio simultaneously doesn't hurt. The accelerator gives you a second knob. It's hard to manage two separate variables with just one knob.

There are two questions asked here.

1. How neutron cross section increased for Thermal neutrons? (As per the title)

Nuclear reaction is just like any other chemical reaction. Just the presence the reactant does not cause reaction. There should be enough time and collisions between reactants. By slowing down the neutrons, other atoms get enough time to capture neutron. In thermal reactor, fast neutrons are slowed down by thousands of times. (When fission happens, released neutrons have around 80% total energy emitted. So Fast neutrons are incredibly fast). This is all just my theory.

2. Few of the neutrons are absorbed by some other atoms which are either don't undergo fission or doesn't emit further neutrons. And even then how more fissile atoms are created in breeder reactors? (As per second paragraph in the content.)

Simple answer - by lot of engineering work. A nuclear reactor is designed around the solution for this problem. For example, Xe is formed as part of fission product. Xe eats lots of neutrons if present in reactor. But in LFTR, Xe comes out of reactor as it forms because Xe is a gas and reactants are in molten state.

You are correct that the average cross sections are much higher in a thermal reactor than in a fast reactor. However, it is not the magnitude of the cross sections that matter, but instead the ratio of the fission to absorption cross sections in the fuel that matter for breeding. This ratio is often referred to as the greek letter "eta" $eta$. (see definition at https://en.wikipedia.org/wiki/Four_factor_formula )

A plot of $eta$ as a function of different isotopes and energies can be seen at https://postimg.cc/7b5gTVVG This plot shows that at higher energies (i.e. fast reactors), the value of eta is much higher than at thermal energies (i.e. thermal reactors). The higher value of eta means that more neutrons are produced per fission, and these extra neutrons can be used to convert fissionable isotopes (such as U-238) to fissile isotopes (such as Pu-239). You need $eta$ values approximately greater than 2.5 to reliably breed more fuel than you consume.