As mentioned in the page previous to the one you are showing, beta is introduced to give nodes with zero in-degree a small amount of centrality.
As the network in your diagram, the node 1, 2 only leads to other nodes, but there is no node that leads to them. They will have zero-centrality if we imagine a random work on the graph (suppose you start at node 1, 2 on the graph, and walk randomly with the arrows. You can never get back to 1 or 2 because there is no arrow leading you back to 1 or 2.). Correspondingly the eigenvalues of the adjacency matrix is always zero.
Beta is introduced to fix that problem by forcing a small amount of centrality to those nodes (like the 1 and 2 in the problem). Since we only care about the relative magnitude of centrality (i.e. we always normalize them), we can simply take beta to be 1.
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I have modified the solution according to your notes.
