EDIT: It occurs to me that I didn't directly talk about twisted objects. The idea is the same. Let us start with an object $Y'$ over $U_0$ equipped with an isomorphism $f_0:Y'\rightarrow Y\times_XU_0$. This isomorphism has to satisfy the co-cycle condition, which means the following:
We have two projections $p_1,p_2:U_0\times_XU_0\rightarrow U_0$. So we have two different ways to pull-back $Y'$ to over $U_0\times_XU_0$, giving us $p_1^*Y'$ and $p_2^*Y'$. The two pull-backs of $Y\times_XU_0$ under these projections are canonically isomorphic, since the two projections $p_1,p_2$, when composed with the map $U_0\rightarrow X$ agree with the structure map for $U_0\times_XU_0\rightarrow X$. So pulling back $f_0$ gives us isomorphisms $$p_1^*Y'\rightarrow Y\times_X(U_0\times_XU_0)\rightarrow p_2^*Y'.$$
This is your $\alpha_{0,0}$; if it satisfies the co-cycle condition--this amounts to the required compatibility between the pull-backs of $\alpha_{0,0}$ to $U_0\times_{X}U_0\times_{X}U_0$ under the three different projections to $U_0\times_XU_0$--then it gives you descent data for $Y'$. In the Galois setting, if you use $f_0$ to identify $Y'$ with $Y\times_XU_0$, then $\alpha_{0,0}$ is giving you a `twisted' action of $G$ on $Y\times_XU_0$.
This data is not always effective. I would highly recommend the chapter on descent in Bosch-Lutkebohmmert-Raynaud's `Neron Models' for an explanation of all these things.
