For me: It determines whether or not I'll buy a console at launch, or wait a year or two for the library to improve. Even if I can afford it, I have a tough rationalizing buying a system purely on the promise of future games. On the other hand, if the new system allows me to play my existing library, I feel much more compelled to purchase the new platform.
Basically...it's easier for me to rationalize buying an upgrade, rather than an entirely new system with a library that may not appeal to me yet.

In general: I don't think it makes a huge difference, if any at all, as the currents sales paths seem to prove.

Don't know if it's overhyped but for some people is a cool feature, nothing wrong. Personally I don't care about it, whenever I buy a new console it's because I want new games, unless they are remakes or remasters which actually improve the original game/ have a great value.

However you can't deny BC helps sometimes, I will eventually get the Wii u and I really need to replay Mario Galaxy 1/2, twilight princess in the future so I was able to sell my Wii without regrets :P

It is when I saw some XOne fans claiming this is what won E3 for MS lol.

It is good features but not really must have for me. Unless you plan to sell your older console and keep the games it is not really big deal. I have original PS3 with full PS2 support but never really use it. If I want to play my PS2 games I will use my PS2.

I've owned several consoles with backwards compatibility, and I literally only ever used it on Wii because of most of Wii's library didn't appeal to me. I've always seen BC as nothing more than a convenience for anyone who doesn't own a last gen system or is too lazy to plug it back in. I buy new hardware to play new games that can only be done on that hardware. If I ever feel the need to play an older game, I'll simply go back to that older hardware. It's just not that big of a deal, in my honest opinion.

As far as the big hullabaloo with XB1 BC, it's a nice feature, but it loses a lot of it's impressiveness because the list of games is so limited. It's clear that Microsoft is more focused on using it as more of a marketing ploy (advertising new add-ons to BC list with gen releases, etc.), since they could've easily outright given the XB1 full BC, thanks to the almost identical architecture. PS4 has to rely on things like PS Now, because it lacks such a benefit. It's quite possible that PS5 will have full BC, so long as Sony sticks to the same architecture as well. Either way, I digress. Just not a serious feature, and certainly not a game changer to anyone besides the most hardcore brand fans.

It is not essential, but it is a nice to have. I have sold most of my X360 games with the exception of the digital ones and a couple of favourites that I simply enjoy to much. At the moment they rarely get played as it means turning on the 360. I WISH the One was BC for these, it isn't a game breaker but it certainly is a very nice feature that previous gen owners appreciate and would perhaps help keep 360 owners loyal.

 One really has to wonder why MS didn't do full on BC right off the bat. It could've been a decent enough marketing tool for them, that would've helped at least attract more 360 users, if nothing else. 

The fact that it now allows them to give away free 360 games with new games means that it is not over hyped at all. Fallout 3 with fallout 4, all the gears games with the remake ect. Figuring out bc has allowed them to easily do this, which is great for the consumer.

It's a neat feature but that's all, and also Xbox One backward compatibility is not a true backward compatibility , and only work for small selected games. and also it's a bit too late for that, backward compatibility very useful if it's on first year of console gen transition, Now people only look and wait for a future games and the upcoming new games. and also Xbox 360 lack of true exclusives unlike PS3.

It's just a very nice and useful feature.

I don't think BC should ever be seen as this uber-tiered feature on any console.