How do I stop a iPad from closing out my tabs?

iPadOS15 permits you to have multiple “instances” of the same App - each instance being self contained. The shelf, when triggered, shows a thumbnail of each running instance of an App. 

For example, if you have three running instances of Safari, each with its own set of open tabs, you will see three thumbnails corresponding to each running instance of Safari - plus a button to open a new instance of Safari. Simply tapping a thumbnail will switch directly to that running instance of Safari. Each App - such as Mail, Safari, Notes, Calendar, etc. - has its own corresponding shelf.

Individual open instances of an App can be closed directly from the Shelf; just swipe the unwanted thumbnail off the top of the screen. Note however, that Multitasking and associated Shelf functions cannot be disabled.

This article from Appleinsider has a useful explanation of this new feature along with many of the updates to multitasking:

https://appleinsider.com/articles/21/08/12/how-to-master-multitasking-on-ipad-and-ipad-pro-in-ipados-15

The iPad User Guide is also a rich source of frequently overlooked information. The Guide is available both online using a web browser (such as Safari), or as an Apple Books download:

iPad User Guide - Apple Support

https://books.apple.com/book/id1567104892

Safari tabs can be configured in Safari Settings to automatically close after one day, one week, one month - or manually - but will not automatically close in a matter of just a few hours.

Settings > Safari > Close Tabs

If your tabs are closing in the timescale that you mention, or at all if set to Manual, then something is either causing Safari to quit without saving its running state, or something else is closing your tabs.

You may need to establish the circumstances in which tabs are seeming closing. We benefit of more information, contributors here may be able to offer more insight or specific guidance.

You mention visibility of the Safari Shelf. Each of the thumbnails represents another open instance of Safari. Have you tried looking through these (just tap on a thumbnail to select it) to see if your apparently disappearing tabs are, in fact, still open in another Safari window?