MacBook Air storage filling up without adding files

That 'gray bar' in your screenshot is not what TheLittles suggests is filling with caches, that is your internal storage. In disk utility, you just selected the snapshot volume that macOS uses during system updates and is part of 'other volumes' when you select the Data volume. Here's mine, the Data volume that's selected is my user data.

If I select the snapshot, I get the picture you posted:

In Disk Utility if you select View Menu > Show All Devices and select the Container on the drive, you get the 'full picture' that shows the snapshot, the Recovery OS (preboot) and the VM swap file volume.

Note that the Free space is essentially constant, that's the real amount of free space on the drive. Yours is showing <10 GB and what that means is that you need to delete data from your internal storage (copy it to an external drive, for example).

Programs like OmniDiskSweeper and Disk Inventory X can show you what's taking up data on your drive.

“space: the disk space is always filling alone without adding anything or files even if I delete files the storage gets filled alone how to fix that”

———-

Mac Storage:

Going into Disk Utility, what do you see is taking up the list of your storage? If there is a large grey line, then caches are being extensively created, and you need to delete them. So, what is taking up most of the grey bar?

“that is the disk utility: how would I delete these”

———-

Thank you for the screenshot.

Clearing the Cache on a Mac:

Unfortunately, on a Mac, there is no way to clear it 100%, without a fresh, new install. There’s lots of was mentioned to perform this, many of which are found in these forums:

How to Clear System Data on Mac - Apple Community

“MacBook Air storage filling up without adding files: the disk space is always filling alone without adding anything or files even if I delete files the storage gets filled alone how to fix that”

———-

Clear your iPhone’s Caches:

Just for the record, you may want to see about clearing the caches on your iPhone —you can do so 100%, through use of a computer. Use my User Tip: How to Clear "System Data" in your iPhone’s Storage - Apple Community

“That 'gray bar' in your screenshot is not what TheLittles suggests is filling with caches, that is your internal storage”

——-

I stand corrected. I did not zoom in on my phone to see the title “Other Volumes”.

Thank you for seeing that.

“that is the disk utility how would I delete these“

———-

My second reply would not apply, being it’s not System Data (caches), formerly called “Other” back in the day. See the reply by @neuroanatomist.