Net connections are the "air wires" that indicate the connectivity of the PCB using thin lines drawn between unrouted points on the board. There are several methods to control the visibility of these connections as outlined below.
Show/Hide Net Connections UI
The user interface provides options to show and hide net connections right down to the individual net level. Launch the viewing panel from View | PCB > Nets.
The nets panel lists your net classes at the top, nets that are members of the selected class next and finally the primitive objects that make up the nets. A quick way to toggle visibility is to click <All Nets>, right click > Connections > Show or Hide. In addition the Show/Hide option is also available on individual nets.
The Change Net Color option in the above menu allows selecting a different colour for the display of net connections and can be applied to individual nets.
The View Configurations dialog can be used to control the visibility of net connections allowing for a global on/off and the default colour. This dialog can be invoked by clicking the coloured box next to LS in the lower left of the PCB editor window.
Single Layer Mode
Single layer mode is configurable to show just the selected layer (tab set at bottom of PCB editor window), grey out other layers or monochrome other layers. Less known is the option to change how net connections are shown.
From the View Configurations dialog select View Options tab you can control if all nets should be shown or just those that have a connection to the current layer.
Another option that may be of interest is the one below - Use Layer Colors For Connection Drawing.
Using the search box (top right) allows control of net connections by component.
This mode activates a selection cursor allowing components to be clicked on to turn on/off their connection lines.
If Connections Are Not Showing Correctly
Try Tools | Netlist > Clean All Nets. This has been known to sort out designs where only some net connections were displayed.