The main exe, aw-qt, comes with an aw-qt.desktop file. But if I just cut-paste it into the Desktop like I would in Windows, it won't work. What should I do?
Your first thought might be to create a symbolic link to the executable. Those can be quite convenient, and are like a better version of Windows shortcuts, but, as you'll see in the linked page, have their own hidden complications.
Your second thought might be to right-click on the desktop menu and see if there's an option to create a 'shortcut'. Turns out there is! Though it's called a "launcher". Clicking there will likely get a GUI to open - a Graphic User Interface, you know, a window with menus and buttons and fields and stuff - I like to pronounce it "gooey".
The GUI will be asking you exactly what command you want the commander to launch, or what app do you want it to use to open which file, or whether you want it to be visually distinguished through a specific icon. I mean inclusive "or" - exclusive "or"s where you can either do one thing or the other but not both are seldom used in my experience, so instead of using "or" for exclusive ors and "and/or" for exclusive ones, I prefer saying OR for the inclusive kind and XOR for the "eXclusive OR".
You could then go to wherever you stored the executable and all its libraries - that's what is usually meant by "the binaries". You'd then get the address of the executable, i.e. its file path, which is likely in the very first folder of the binaries of the app, the one that contains them all, the app's root folder (not to be confused with the OS's root folder!). In the case of ActivityWatch, it's called aw-qt. Just that, no .exe! Unlike Windows, Linux systems usually don't need the file's name to tell them what kind of file it is! When a file ends with, say, .md or .desktop, it's mostly for the user's convenience.
You could also, if you want, look for the app's specific icon: just find the search function of your file manager, and ask it to look for all images within all the folders and subfolders. It that icon exists, it's most likely there. Get its address, and input it in the appropriate filed in the launcher-creator GUI.
Now, here's the thing, though: your self-created launcher will likely be missing some features that the devs intended and wrote into their original .desktop launcher. So here's what you can do: just open them with a text editor! Those things are like Notepad in Windows, but waaaay better. You've got your vim, your EMACS, your katia, your gedit, your notepad++... they'll likely be easy to find on your Start menu under "editor," "development," or "programming", be it a folder name or a search term.
If for some unfathomable reason your installation doesn't have one at hand immediately, you can also find them easily in your app or package manager or installer, under those very same search terms. Get one! They're extremely useful. Though if you're at the level where my advice until now doesn't make you roll your eyes, I suggest you avoid Vim - that thing is powerful, but has a learning curve. It's kind of the Dwarf Fortress of text editors. Emacs is more like the Starcraft. If you don't know about either, bless your soul, don't even worry about it.
Now, make sure you've cut the original .desktop that came with the app, and pasted it to the desktop. Right-click on it, and select "open with...", and open it with a text editor. You'll see the options laid out, in very plain text, before you. Where it says "Exec=, write the proper, full path to the original executable, which you can find in its Properties, among other ways. Don't forget to end the path with the executable's own name, in this case aw-qt!
Likewise, where it says Icon=, put the whole path to the icon, in this case named logo.png.
Save the file, close the editor, double-click on the app, and watch as it runs like Forrest Gump in a Midwestern prairie road. Enjoy your app!