To simplify when using iTunes Match, your phone or other device, connected to the internet, can play "your" music from Apple's Servers. Apple scans your physical music Library and then "Matches" the songs to songs they already have.
This means that if Apple has a song on their server that "Matches" a song in your physical Hard Drive music library then it can stream if from the Apple server to your other devices instead of streaming it from your actual computer cuz that would be rot with issues unless setup properly.
The caveat is... that if you have songs not on Apple's servers i.e., mixtapes, live recordings, rare CD's etc. Then Apple will upload those songs to it's Servers to stream to your other devices but there is a limitation. (read the fine print)
Now, iTunes Match offers three main features:
Matching my local collection with Apple's, thus giving me (potentially) better encoding quality.
No. (or yes) The quality of streaming from Apple Server is 256kb. That's not good quality and has nothing to do with your physical song library. (or yes) if you have really bad quality rips that are below 256kb (yuck)
Making music available through the cloud, making it shareable with my family (which would otherwise be trivially replicated by putting music files into other peoples' iTunes collection).
Correct! (kinda) Your family will have to use your Account either in iTunes or "iTunes & App Stores" (read the update at end of post) If you have a song in your library and it "Matches" a song Apple has on it's server or Apple Music library then your family can listen to that song streamed from Apple's Servers. Be careful though because music on the other devices is not actually "yours" it's Apple's and when you cancel your subscription that music will be deleted from any device that was not the original database (your mac with the real physical files)
Making storage management on iOS devices easier by pulling songs from the cloud only when needed.
Correct (in a sense) You are streaming the song not downloading it unless you choose to download the song.
With an iTunes Match subscription, iTunes matches your library, uploads any unmatched tracks (limited upload) and makes your music available on multiple devices through the iCloud Music Library. If you have low-quality tracks, iTunes Match “upgrades” them, allowing you to download 256 kbps AAC files without DRM. You can also download your files to any computer linked to your iTunes Match account, and listen to your music on any linked iOS device. iTunes Match has a limit of 25,000 tracks, not including iTunes Store purchases.
However, there is one essential difference between iTunes Match and Apple Music. While iTunes Match lets you download your music files and then play them anywhere, Apple Music adds DRM to your files. This means that if you rip a CD, and it’s matched or uploaded to iCloud Music Library via Apple Music, and you download the files, say, on another Mac, you will only be able to play those files as long as your Apple Music subscription is active. If you delete your originals, or lose them, then you won’t be able to access files without DRM. As such, it is essential that you keep backups of your original files if you use Apple Music.
If you have both an iTunes Match subscription ($25 per year) and an Apple Music membership ($10 per month), then you get files without DRM.
The problem is that Apple is not making a distinction between Apple Music files that you download for offline listening—this is a key feature of Apple Music—and files that belong to you, which are matched or uploaded. In addition, there seem to be bugs right now, causing many previous matched files to show as Apple Music files, and to contain DRM when downloaded, even for users with iTunes Match subscriptions.
UPDATE: As mentioned in the comments below I wanted to add that iTunes Match cannot be shared to other Apple/iTunes/iCloud ID's. Unlike a "Family" Apple music subscription iTunes Match can only be associated with one Account. This does not mean that your "family" or other devices you and your family own cannot take advantage of your iTunes Match. Those devices would just need to put in your Account info into iTunes or in "iTunes and App Store" (iOS). On a computer this is not so bad as you can have iCloud syncing/backup/App Store separate from iTunes (not sure about the new iTunes in Catilina though)
On a mobile device your App Store would be linked to the iTunes Match account unfortunately. So purchases on those mobile devices done by your family would be on just one account. Maybe not so bad but it kinda negates the family sharing aspect of iCloud only in the sense that you won't have separate App store accounts so everything everyone purchases will show up on the other devices unless you deselect "Apps" in "iTunes & App Stores" which would turn off syncing across all devices. That's not so bad but if you have multiple iOS devices (iPhone,iPad,iPad mini) then you'd need to go into the App store on the device that did not literally download the App and then go into purchases or search for the App to manually download it to that device. How you set this up is up to your and your family.
The draw back being that if your "wife" made a playlist on her iPhone then you would see that on your device and visa versa. If you wanted to get tricky you could create separate itunes libraries (Computer only) by holding down the shift key when opening iTunes. Then if you wanted, your family members could have their own iTunes minus your iTunes Match songs and just need to hold down the "shift key" when opening iTunes to select the Library they want to use. Not a perfect solution but I just wanted to give you some real world examples that I hope you can use. ;)