I have not cashed in my free account yet (because I have a ton of my own music , and don't need to play through the Alexa devices I can easily stream my entire digital collection straight to my stereo, or to echo devices via BT, but anyway, I think I understand it, and basically the more you pay, the more you get.
Prime members can get free unlimited for ONE device playing at a time for the lowest price of $3.99 (I believe we can switch device, but probably is some sort of hassle through the music account or maybe through Alexa app.) Non Prime members pay a few dollars more for same. Anyone can get unlimited for all their devices to play for around $9.99. All prices are per month. It is all explained at the Music Unlimited section at the website.
The one month free offer may be for the top tier premium plan, which dissolves after a month, and you need to pick one of the plans if you want to stay connected.