If you want to stay on SketchUp, you could try an animation extension such as Keyframe Animation. You would have to setup the whole assembly manually exploded, afterwards frame each step, and finally export the animation from the file menu.
If you want something better suited for the particular task, consider Solid Edge.
Solid Edge is capable of creating such animations by combining the Explode, Animation and Render tools.
Solid Edge is a CAD software that focuses in the design of machines and mechanical parts. You would need to export the models in a format that Solid Edge supports (there are extensions to do the export - which I'm not mentioning because I can't test them right now - you may also use Blender to import SketchUp and export for Solid Edge).
Alternatively, you can recreate them. After all, SketchUp has already taught you the basics of CAD, I assure you will find Solid Edge easier than more traditional CAD solutions such as AutoCAD. This is in part to its "Synchronous Technology", which I would argue influenced SketchUp.
Using Solid Edge, what you do is as follows:
- Auto explode the assembly.
- Move the pieces to where you want to them to start, and out of the way of other pieces.
- Open the animation tools (which automatically takes adds the previous steps you did) and adjust the timings for the animation, and add any other effects you want.
- Render the animation to video.
Disclaimer: It has been more than a decade since I last used Solid Edge, so take my description of the steps with a grain of salt.
I found these PDFs that will help you with the details: ERA 1, 2.
Alternatively look for video on Youtube, such as this one: Explode Render Animate.
You can look for video to gif solutions if you want a gif. I do not expect them to have added support to export to gif directly.