I-tunes downloads
How do I download music to my iPhone from my Home sharing library? I don't think I can access my Home Sharing library unless I'm in the same area as my iMac.
iPhone XR, iOS 15
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You don't. Home Sharing just allows you to share music from your computer's iTunes Library to your devices while on the same Wifi network.
If you want to have the music directly on your device when offline, then you can use either the iCloud Music Library option from the Apple Music subscription service if you are paying for it, or you can simply directly sync your iPhone to your computer to transfer the songs directly over USB or Wifi as necessary.
Sync your iPhone, iPad, or iPod touch with iTunes on your computer using USB - Apple Support