Surprisingly for me, this task is about flows.

Let's create a graph with source($$$S$$$), sink($$$T$$$), vertices representing vertices of the original graph($$$v$$$), and vertices representing edges of the original graph($$$e_{uv}$$$). Let's connect $$$S$$$ with every vertex $$$e_{uv}$$$ with an edge of capacity $$$1$$$, connect $$$e_{uv}$$$ with $$$u$$$ and $$$v$$$ with edges of capacity $$$1$$$, and finally connect $$$v$$$ with $$$T$$$ with an edge of capacity $$$d_v$$$. Claim is that now we need to check that maxflow in this network equals to the number of edges.

The trick with these vertices representing edges works in a way that one unit of liquid which came to $$$e_{uv}$$$ goes to either $$$u$$$ or $$$v$$$ (because in a network with integer capacities there's a maxflow with integer flows through each edge), so that's how we pick an orientation of an edge. Edges to sink with capacities $$$d_v$$$ guarantee that every out degree does not exceed $$$d_v$$$, so, to check that we succesfully oriented all edges, we may check that maxflow in this network equals to the number of edges.

For some of you, it may seem too clear. If that's the case, sorry for your time. I find it very challenging to reduce a problem to maxflow, so if you have any insights how to learn it or know some other tricks, please feel free to share them. Thanks :-)