The simplest explanation of how to do it is that you want to replace one of the "wires" with your meter. This way the current flows through the meter on it's way between the power source and the load.
You have to make sure the meter leads are plugged in to the correct jack. Usually, there is a common or ground jack used for all measurements. Then you will usually have at least two positive lead jacks. One will normally be for volts and ohms, and the other will be for current. Sometimes there may be an extra jack for very high voltage. But more commonly, you will find extra jacks for different current ranges. Since the meter acts like a short when in use, it will usually be protected by a fuse to prevent damaging the circuit or harming the user if there is too much current. You need to make sure the current you want to measure, the range selected on the meter, and the jack used for the positive lead, all are matched correctly. If you are unsure of the current, start at the highest range and work your way down if needed.
CAUTION: Make sure you do not try to measure voltage while the meter leads are in the current jacks. When measuring current, the meter has very low resistance. So if you put just the meter across even fairly low voltages, it will let too much current flow and blow the fuse. And yes, sooner or later we all get distracted and do it. Buy spare fuses for your meter.
You don't actually need to replace a wire, you just need a "break" in the electrical path where you can insert your meter. For example, if you are just trying to measure how much current is being drawn from the battery of a flashlight, the simple way is to remove the tailcap. Most lights make the negative connection to the battery via the tailcap. So you just remove the tailcap, turn the switch on, then put your leads on the back of the battery and the light body. The meter is now completing the electrical circuit instead of the tailcap, and the meter will display the current flowing.
Measuring the current flowing to an LED when using a driver circuit can be more difficult. First, you'll usually have to desolder a connection in order to be able to insert your meter in the path to the LED. Also, this can cause additional problems since there are some drivers that can be damaged if you apply power while LED is not connected. In that case, make sure you have the meter fully in the circuit before you apply power.
