Why both boost convertor and boost controller are required here?
The schematic for that reference design makes it pretty clear.
The TPS63061 boost converter provides a boosted, low-current supply for the TPS43061 boost controller's internals to run from.  

That circuit has been designed to be able to run from a single Li-Ion cell, but the TPS43061 controller requires at least a 4.5V supply - which a single-cell can't provide.  

But even though the controller is running its internals fro the boosted supply, it is controlling its FET which is running directly from the battery.  

So the TPS63061 is only supplying the TPS43061, while the TPS43061 controls the FET which supplies the output.  

If you don't require the single-cell functionality, you can remove the TPS63061 and connect the TPS43061's Vin pin directly to the input supply.