The text which I was reading said that if we assume points $\displaystyle O_{1} ,O_{2} ,A,B,C,D$ on a circle then it's cross ratio is $\displaystyle R( O_{1} A,O_{1} B;O_{1} C,O_{1} D)=R( O_{2} A,O_{2} B;O_{2} C,O_{2} D)$
please explain what does the above mean(by expressing in ratio) and give its proof.
The cross ratios you are giving are cross ratios of pencils of four lines (four lines passing through the same point). The definition of the cross ratio of lines $(L_1,L_2,L_3,L_4)$ is based on the theorem saying that whatever the line cutting the pencil, the (ordinary) cross ratio of intersection points $(I_1,I_2,I_3,I_4)$ has always the same value.
See for example this reference.
The value of the cross ratio of 4 lines can also be computed "intrinsically" by using angles as given for example here, with their notations:
$$(\sin(cMa)/\sin(cMb))/(\sin(dMa)/\sin(dMb))$$
Consider the following figure (using your notations):
The two cross ratios are the same because the four angles involved in each one are the same (and therefore their sines as well). This comes from the classical theorem saying that, in a circle, inscribed angles subtended by the same arc are equal (for example the two angles $\alpha$ are equal because they are both subtended by arc $AB$).