Fuses in Wires.... Yes... No....

Better yet you can use inexpensive breakers and add the ability to easily shut down your system for service. Square D QO and QOU breakers are rated for low voltage DC. Midnight Solar breakers should be used on higher voltage systems, up to 300 volts.

Are you asking 'How do you physically connect the fuses to the wire or cable?"

If so it depends what type of fuses or breakers you're going to use.

You can either use two separate cables or cut the one you have in half as long as its long enough.
Many breakers simply require you to insert the bare end of the cable into one end then you secure it in by tightening the little clamp with a screw driver or similar. Probably much the same way you connect your cable or wire to your Solar Charge controller. Same with the other end.

Fuses generally need a fuse holder, which are usually purchased separately or may come with your fuse. How you connect the fuse holder to the wire can be many and varied. Some fuse holders have a screw and clamp system like the breakers. Others may just require crimping (with a crimping tool) with various end connectors (rings or spades etc) or simply the bare wire twisted and joined with standard wire connectors. (An plastic insulated screw and clamp)

As to where you physically put them, depends on many things but its generally best to put the one connected to the battery closest to the battery terminal but somewhere easily accessible.

Same with the inverter etc. In small one panel systems the one from the solar panel is not so important and sometimes not even included, but recommended so the panel can be isolated from the controller if need be.

As to the size etc it will depend on the current draw. They are usually rated in Amps, for example a 30A rated fuse will usually pop when more than 30a is going through it. (and I'm being very general here.) Each of these fuses or breakers has to be sized to the current draw on that particular part of the solar system. For example the fuse or breaker between the solar panel and control woud be rated to slightly more current (amps) that your panel(s) produces (Isc). I think the diagram you attached says Isc x 1.2. And not more than the rated current of your charge controller as well. Each individuall cable needs to be rated to also carry that current. The Isc of your solar panel can be found on the back on the information panel. In general for a single panel low wattage panels these cables are relatively small and so are the fuses.

The one between the battery and controller is usually rated the same way or simply around the maximum of your Charge Controller and cable. Again being very general Controller rated at 15amps, a fuse or breaker rated at the same along with a cable or wire rated to carry the same.

The one between the inverter needs to be rated to the inverter power rating and will generally be much larger along with heavier thicker cables etc.

The fuses are there to protect the components of the solar system from being destroyed or damaged in the event something goes wrong and also to prevent cables, wires or components melting or catching fire in the event something goes wrong. The fuse pops or the breaker, breaks the current before that event can occur. (hopefully)

Remember I'm being very general here just to give you a simple idea. Other posts here explain how to calculate what you need depending on the specifications your system and its components. .