If you wanna go down the rabbit hole, read at least the first few chapters of Mastering Lightning

https://github.com/lnbook/lnbook

Another great resource I have learnt a lot from is the voltage blog. Many well written and easy to understand articles the Lightning topics I have had questions for.

This post discusses multipart payments which your question is about.

https://www.voltage.cloud/blog/multipart-payments-on-lightning-network-break-down-amp-and-mmp

And this may be a good place to start learning about running your own personal node in general.

https://www.voltage.cloud/blog/lightning-node-type-series-1-the-consumer-node-or-how-to-live-on-lightning

Not sure how far along the journey you are but I have found this my goto site for learning something about lightning.