What you need is a video editing program. Macs can use iMovie. Windows users can download
Windows Movie Maker from Microsoft. Take a look at the article at Digital Trends for more free options:
Best Free Video Editing Software
Even the free programs let you edit, add titles, music and transitions. But all you need to do is the first function - editing. So it's not that hard. You only have to learn to do one thing.
The steps are pretty much the same in any program.
Open the program and start a new movie - this is a new blank file; the new file is usually called a "project".
Add your downloaded video into the project. If all you want to do is get to one part of a video, that's the end of this step for you. But usually at this point you are adding multiple videos so you can combine them together.
"Trim" your video - that means you are going to set a "start point" where you want the scene to begin, and an "end point" where you want it to stop.
Another way to think of this is to picture the old days when movie editors actually worked with physical film in the editing process. If I wanted to cut out just one scene in a longer movie I would take my razor and make the first cut where I want to start (the start point) and then forward the movie to where I want the scene to end and make my second cut (the end point). I could then hold in my hands one long strand of film that had my scene (and throw away the rest).
Now to get your final product you have either one step or two, depending on the program.
Save your "project" file (that records where you made your edits).
Generate a new movie that is now cut to your specifications. Technically this is called "rendering", but most consumer-oriented programs avoid that term. And some programs combine the project saving with creating a new movie in a single step.
You are done!
One limitation you may run into is that consumer video editing programs can be limited in the video formats (or "codecs") that they can use. You might also need install a new codec on your computer (like DivX) that knows how to handle a specific type of file. But, just forget I even mentioned that; at that point it's not worth the effort unless you are going to do this a lot.
Unfortunately you can't always know what codec is used in the movie, because the video is enclosed in a "wrapper" - which is the file extension that you see (.avi, .mp4, .mpv, for example). So you just have to try it out and see if it works for your video. But, what the heck - the video editing program is free.
One solution around these issues are to convert the downloaded video from one format to another one that your editing program can handle. For that, I recommend downloading Handbrake if you are a Windows user.
Ok - go to it!