STV vs. IRV
The main changes that I've read about were that it would introduce instant-runoff style ranked choice voting and that it would put districting in the hands of an independent commission that would create much larger districts with multiple representatives versus single representative districts that currently exist.
In a multiple representation district, the voting method is called Single Transferable Vote (STV) rather than Instant-Runoff Voting (IRV). It's also noteworthy that voting can be STV in multiple ways, including Condorcet-compliant versions.
The best part of the proposed Fair Representation Act is that it would radically reduce redistricting angst. In many states, there would be no redistricting, because there would only be one district. And because each district in a multiple district state would have multiple representatives, people could choose for whom to vote. Don't like the traditional Republican who represented your original district? Vote for the minority or Sanders candidate instead. If there are enough of you across the district, you can get your representative.
It doesn't entirely eliminate redistricting like statewide districts do, except where it converts to statewide districts, like Rhode Island and Connecticut. What it does do is make redistricting less impactful. Even if you pack the districts with one party or dilute one party across multiple districts, having multiple representatives per district means that it doesn't matter.
An example. In Iowa, there are four congressional districts. If you shifted 28,000 Democrats (net) from IA-2 to IA-4, all four would have been Republican after 2016. Or if you shifted a similar amount of Democrats from IA-2 to IA-3 in 2012, three out of four would have been Democratic. But if all four are chosen in an STV system, the most likely result would be two and two. Perhaps in an extreme wave year, three to one (as it is now after 2016).
Roughly 775,000 voters picked the Republican and 675,000 voted for Democrats. Assuming that the Republican voters would vote for all four Republicans and the Democrats would vote for the four Democrats, this would give two Republicans and two Democrats.
The reason that this works is that it allows voters to choose how they group. Voters can group geographically, like in the current system, voting for the hometown candidate. Or voters can group ideologically or ethnically. Under the current system, politicians choose how voters are grouped. In this system, voters would choose for themselves.
Radicals vs. Moderates
The other answer asserts that this will help elect small party candidates and radicals. I don't know that either are true. First, five-seat districts wouldn't be enough to elect any of the third parties. They get 1-3% of the vote, not the 15-30% it would take to win a seat. Second, this underestimates the ability of the current system to elect radicals.
Now, when a moderate runs against a more partisan candidate, the more partisan candidate frequently wins. Eric Cantor, John Kasich, Tim Holden, and Joe Lieberman all lost partisan races to more radical candidates who then went on to win the general election. Now, it's true that these more radical candidates would still have a constituency. But when voting in lists, so do the moderates. Because moderate voters can get their choice. The Democrats who would prefer John Kasich to Donald Trump or Eric Cantor to David Brat. Or the Republicans who would prefer Tim Holden or Joe Lieberman to their opponents.
Under the current system, such candidates don't even make it to the general election, even though a majority who will vote in the election would prefer them to one or the other candidate.
As the electorate becomes increasingly polarized, we need to do something to allow at least some moderate swing voters in to vote on legislation. Otherwise Congress is grinding to a halt. Republicans can't agree on increased taxes and Democrats can't agree on reduced spending. Because the partisans are taking over, there is no middle ground left for compromise.
It's also not entirely true that third parties oppose compromise. If they did somehow get through this system (and it's possible that their vote share would swell under the new system), they do have some room for agreement with the main parties. Libertarians could team with Republicans on taxes and spending or team with Democrats on marijuana legalization. Greens are less bipartisan but have overlap with the Democrats.
Currently, almost every Republican is from a suburban or rural district. Almost every Democrat is from an urban district. What about urban Republicans or rural Democrats? They have no representation for their views. This makes it even harder to compromise, as House Democrats have no knowledge of issues affecting rural white folk and Republicans have no knowledge of urban voters.
This is because Republicans in urban areas have no input into candidates in their areas. The Democrat always wins, no matter who the Republican choice is. The other way around for rural Democrats. This system would change that. A four-member district means that even someone only representing 25% of the voters will have a seat. The party that represent 75% of the voters will have three seats.
Overall, I would prefer no districts below the state level. That would allow for true third-party candidates in the larger states like California, Texas, Florida, and New York. This proposal leaves the two-party duopoly intact, although it allows for the possibility of a real third-party to develop. Under current voting patterns, Democrats and Republicans would win all of the seats. But it would be possible for a party like the Bull Moose or United We Stand parties to develop and win some seats. If neither side holds a majority, then Republicans and Democrats would have to learn to work with other parties to pass legislation.
Third parties change the dynamic as well. Negative advertising doesn't work as well. Negative advertising tends to hurt the advertiser as well. It's just that it hurts the target more. In a two candidate election, negative advertising against your single opponent means that you do better in comparison. In a three-way race, it may mean that your other opponent picks up votes from both your leading opponent and you.
It also makes it less clear who your opponent actually is. Is it someone from the other party? Or one of the others from your own party? More importantly, there is more room to compete for the middle.
I would prefer a Condorcet-compliant method like Schulze-STV to reduce tactical voting. This proposal uses traditional STV rules. But just having multiple winners per district reduces the impact of tactical voting. And collecting the ranked votes makes it possible to score them however.
I'm not crazy about "independent" commissions selecting the districts. In practice, the independent commissions still produce slanted districts. This goes back again to why I would prefer statewide districts. Then voters choose how to group rather than someone else choosing groupings for voters.