Step by step is going to be challenging, but possible.

1. Utilize chain tool to break current chain
2. Remove current derailleur from derailleur hanger.
3. Install similar model shimano rear derailleur
4. Route chain, remove the rest of the outer link broken to remove the chain, install master link

5. Route derailleur cable, set high limit screw, and secure cable pinch.

6. Tune cable tension and set low limit screw.

Rear derailleurs don’t often fail where the solution is just to replace it. If it went into the wheel and was damaged. The wheel, hanger, and chain may need service.

Utilize Park Tools YouTube channel to give a better visual representation. Similar to many YouTube solutions it will not be your exact item, but the resolution may be similar.

If you are buying parts anyways, bike shop labor would only be $40 max in my shop in Minnesota. It might be worth the time/money to have a shop give you a quote. With parts you could be close close just a new bike of similar quality

The best option if available is to get help from a bike co-op or at a bike shop's repair clinic. If not, just Google "install rear derailleur". IF your derailleur was damaged by going into the spokes it is critical to have a bike shop look at the frame where the derailleur is attached.

I had a look at the Mongoose website.
They have manual for download.

https://int.mongoose.com/pages/product-manuals

Enter the model of your bike, or look for a picture of it.
Follow the instructions....

REgards
Geoff