Why has the auto-router put my PCB traces on two different layers?
It appears that your autorouter is using one layer for north-south traces (red) and another for east-west traces (blue).

Figure 1. This side of the board is almost all north-south routing. The east-west routing will be done on the other side. Image source: Advances in printed circuit boards.
This is a sensible strategy as it is an easy design rule for a human or autorouter to follow. It also means that future additional north-south traces won't run into obstructions that require re-routing or circumnavigation.
Regarding placement advice:

Some components such as edge connectors, push-buttons, LEDs, displays, etc. will have pre-determined positions to line up with enclosure holes. Position these first.
If manual assembly is to be used then consider component orientation. Don't, for example, put one of ten chips rotated 180° from all the others.
The schematic should give you a good indication of what components need to be grouped in close proximity due to the number of traces between them. Work on these as a sub-section of the PCB.
Certain components will have recommended PCB layouts on their datasheets and these have to be followed to ensure good performance.