Save music from itunes to iphone locally.

I'll assume you have Windows 7 or later, which is a requirement.

Open iTunes and connect your device to your computer with a USB cable.

Click the device icon in the upper-left corner of the iTunes window. Learn what to do if you don't see the icon. 

From the list under Settings on the left side of the iTunes window, click the type of content that you want to sync [copy]  To turn syncing on for a content type, select the checkbox next to Sync.

Select the checkbox next to each item that you want to sync.

Click the Apply button in the lower-right corner of the window. If syncing doesn't start automatically, click the Sync button.

This will copy all of the music in iTunes to your iPhone. If you don't want all of it click Selected playlists, artists, albums and genres. It will then display all of your music and you can select what you want to copy to your iPhone.

See this article: https://support.apple.com/en-us/HT201253

The link provides detailed instructions. But they are different depending on what kind of computer you have. That's why there are 2 choices when you click the link. If you post your type of computer and operating system I will be happy to open the link then select and copy the instructions for your computer and paste them in this message thread.

Oh, and downloading music from a computer to an iPhone is called "syncing".