Songs are a big part of fantasy literature, especially where Tolkien is concerned. Setting aside whether you like Tolkien's verse, a Middle-earth game must have rules for songs to be complete.
The problem is that RPGs have historically done a terrible job with songs. D&D set the tone early, with bards being relegated to bland buffbots. It wasn't that the rules for songs in D&D were uninteresting (though they were). It's that there was no interesting fiction attached to them. For repositories of oral lore, bards sure weren't encouraged to keep track of what their songs were about.
This has continued right up until the modern day. Songs give buffs (TOR does this; more in a moment) and that's it. You can track your songs or write something in-character about your adventures, but there are scant few hooks to get you to do so, just like in the old days.
TOR was in a similar boat. For being in a world where song is so important, the base rules relegate it to a social skill only. Rivendell fixes this and lashes it to the fiction.
The PCs can now write songs as a Fellowship activity. The process is simple: roll Song to compose it, with the success of the roll determining how difficult it is to actually sing. You then (and this is the important bit) log the details of the song in a communal songbook for reference throughout the campaign. The idea is to create a shared history through the songs the characters write. It's simple, and by no means does it have to be the emphasis of a campaign, but the mere existence of the songbook tells everyone at the table that songs are important.
Each song is of a particular type and the difficulty is altered during the writing, with Traditional and Thematic being harder to write, Elvish more difficult. The song types break down as: Traditional (tied to a specific culture, +2 difficulty for outsiders to sing), Thematic (cannot be sung outside a specific event, like meal-time or marching), or Elvish (harder to write but can be used more often).
So what are the benefits? Like most rules in TOR, there's a theme of leaning on and reinforcing your friendship in the song rules. The party can sing each song in the songbook once per adventure (twice for Elvish). If you succeed, you're Inspired, giving you two extra dice to spend on rolls for the duration of the adventure.
This can be game-changing in narrow circumstances, like a hairy combat roll, but it's not remotely overpowered. And this is exactly what it should be: a nice benefit without being too much, with a deep attachment to the fiction.
It's this attachment to the fiction that TOR does so well. Because the designers seem to think in terms of discrete and self-contained actions (a la a board game, as stated in the last edition of this quick look at Rivendell), they can drum up the "feel" of Middle-earth in very little text.