Congratulations on a useful piece of kit. Logic probes seem to be a disappearing tool these days, but they are still very much worth having on your work bench.
Connect the two crocodile clips to the power supply of the circuit you want to test - red to positive, black to negative or ground. Don't connect to more than about 15V though. Generally any logic circuit will be much less than that.
Then, the probe tip can be touched to any terminal, integrated circuit pin, etc. that you want to check the logic level of.
Logic probes do not tell you a voltage, like a multimeter, and they are not a tool for using on an analogue circuit with op-amps or transistors (usually). Instead they work on a logic circuit (with TTL or CMOS logic gate chips) and tell you if the level is a logic high (1) or a logic low (0). That is above about 70% of the supply for a high, or below 30% of the supply for a low. If you connect to a rapidly alternating pin, the lights will flash.
Note that some pins do not like to be probed, for example ones with a crystal oscillator. That will usually crash the board. Just reset the power then. Also, fast or very low voltage logic circuits might not like being probed (like a PC motherboard for instance), and the probe might not work on them anyway. That's why they are not used as much as they used to be when things were simpler and slower.
Edit to add: Just a cautionary note: Make sure not to slip with the probe and short two pins together. Also, make sure that the crocodile clips don't slip and do the same. I once blew an old computer up when that happened to me and shorted the power rail I had clipped on to, to a couple of signal lines right next to it