I found a way of doing this, but I'm underwhelmed by the solution so hoping Andreas or someone else can weigh back in here, and ideally say if there's a purely olca-ipc method for doing this or another easy way through the GUI. An added "Expand all" command in the Total requirements section would be helpful in this case.
I guess to start with, you can get the total requirements using the olca-ipc using get_total_requirements. However, I was also interested getting the destination for each technosphere flow, like you can see if you were to expand the arrows in Andreas' screenshot (i.e., what processes does the straw from rye production connect to and in what amount). I think there may be way to do this via olca-ipc and upstream trees, but that seemed more trouble than it's worth.
My solution was to export the matrices using "Export as matrix" for my product system (I used excel), but this doesn't provide you with the final solved (scaled) flows. To get the scaling factors, use olca-ipc get_scaling_factors. Multiply these scaling factors along each row of the A matrix, and you should end up with the solved matrix, which will show the total requirements and the processes that each flow is connected to. In Excel, the output flows are in the rows while the destination processes are in the columns and should show as negative values.