Lets get things straight.
Gravity waves travel at the speed of Light waves.
Gravity waves are different than gravitational waves. It is gravitational waves you must mean.
They are disturbances in the curvature of spacetime, generated by accelerated masses, that propagate as waves outward from their source at the speed of light.
So then, why does Light waves get caught by Gravity waves (eg. Black holes) ?
Black holes are not gravity waves ( see link above for definition of gravity waves)
They do not emit gravitational waves unless accelerating, as happens with the merging of black holes.
And if it is about the strength of Photon field,
Photons are elementary particles in the standard model of particle physics. The photon field is part of the quantum field theory that describes elementary particle interaction.
then why are Light waves from a stronger light source (eg. Quasar) still caught by gravity of Black holes on the way ?
If by "caught by gravity" you mean the gravitational lensing observed on light from stars passing close to heavy masses, it is explained by General Relativity.
In general relativity, light follows the curvature of spacetime, hence when light passes around a massive object, it is bent. This means that the light from an object on the other side will be bent towards an observer's eye, just like an ordinary lens. In General Relativity the speed of light depends on the gravitational potential (aka the metric) and this bending can be viewed as a consequence of the light traveling along a gradient in light speed.
You further ask:
Why does Light get caught by Gravity, when both are travelling at Speed of Light?
Gravity is not traveling at the speed of light. If the final quantization of gravity has gravitons, still the force of gravity would depend on the exchange of virtual gravitons, whereas the photons in the light are real particles interacting with the gravitational field ,so gravity is not traveling at any speed. Light does not get caught, it interacts with the gravitational field it finds on the way, which field, in terms of General Relativity, defines the curvature of space time.