Click on the " position " in the Timeline " .. i mean ,, Make sure its Highlighted
That's not a very effective tutorial and there is a much better workflow to accomplish the same thing. If you are new to AE you should take the time to vet your instructors. Most of the YouTube tutorials you find with a Google search are not very good.
That said, if you can't see or select the points on your motion path then you probably should open your Preferences and select Show All Keyframes from the Display Options:
You also want to have the selection tool (v) selected or it will not work. For a little more advanced manipulation of a motion path you can choose the Pen tool (g) and then use the Ctrl/Cmnd and Alt/Option keys to manipulate the bezier controls.
I would suggest using the Search Help field at the top right corner of AE to research things like "edit motion path". You'll find a lot better information than you will with a google search. The Community resources are at least filtered a bit. Assorted animation tools in After Effects
To learn how to use Particular from someone that knows how to use it check out the Red Giant website - They have excellent training that will explain the use of their products in more detail so you can do a better job in less time and create your own designs rather than just copying someone else. Red Giant | Tutorials You can set up your own training library there.
Here's how I would go about setting up the effects used in the tutorial you referred to:
The background would have been made with a shape layer filled with a gradient - they render faster and are more versatile by far than the really old 4 color grid.
I would have added a layer and used the pen tool to create a path I can see on the solid for my lights to follow.
I would have just added the first point light and named it emitter then duplicated the light (Ctrl/Cmnd + d)
I would have set a mask keyframe for the path I drew on the solid layer by selecting the layer and pressing Alt/Option + m then Ctrl/Cmnd + c, then selected the first light and set a position keyframe by pressing Alt/Option +p then Ctrl/Cmnd + v to paste the bezier path to the light. I would have then added a 3D null to the composition made it the parent of the first light and adjusted the position and rotation of the path to give the light a more interesting path.
The procedure would have been repeated for the second mask path and the second light, then parenting removed from the 3D null then I would have repositioned the null, made it the parent of light 2 and then adjusted the path by manipulating the null.
Then I would have added Particular to the Solid I used to draw the motion paths and use the Particle builder to set up my particles. There are a bunch of presets and this is by far the easiest way to get started with your particles. Particles made I would have gone back to the Particular Effects control panel assigned the emitter to the lights and then revealed the position keyframes in the timeline and adjusted the timing by simply dragging the last keyframe for each position to a new time.
Step 1 done in less time with better techniques that show you a workflow that will lead you to something you can polish to your own design in just moments.
I hope this helps. Dropbox - particle trails.aep (your browser may add a .txt extension to the app - just delete it)
Thank you for helping me both of you. I will do as you advise me when I have a little bit more free time.