Common-use kanji (常用漢字) are the characters everybody learns in primary and high-school, as decided by the ministry of culture and education (and science, and technology and - for whatever reason - sports). There's around 2000 characters total - 2136 with this change - and an additional few hundred used for family and given names.
This list matters, as official documents and announcements, school materials and so on can use only characters on this list1. Private publishers can, and do, use whatever kanji they like of course, but it's generally a good idea to heed this list as your readers are guaranteed to know it.
Once upon a time the intention was to gradually abolish kanji. The common-use kanji list was a step in this direction, and newspapers and other publishers were required to stick to this list. But you can't dictate language use from above of course; just look at the futility of French governments trying to keep their language pure of Anglicisms for an example. Languages belong to its users, and it's the users that decide on use, not officials or language academies.
The education ministry no longer tries to dictate use; changes to this list are simply adapting to actual changes in real-world usage. The revision this time around is the largest increase in common-use kanji ever. The reason for the increase is of course that people are using more kanji than before (so much for abolishing them). And the reason for that, in turn, is the computer and the phone.
It's hard to remember how to write a character you rarely use. But it's much, much easier to recognize it when you see it. With a computer or phone you no longer need to remember exactly how to write them - you write the sound of each word and select the appropriate characters from a pop-up list. And there's a network effect: people start using rare characters in email and text, making them more common and encouraging other people to start using them too.
Is this automation bad? Nope. A decent analogy is English spelling: you no longer have to remember how to spell
There are two kinds of additions: Kanji used in place names, and kanji already in common use in newspapers, advertisements and so on.
When I looked at the list itself I was surprised, frankly. Even I know a number of them, and I'd assumed many of them were common-use already. 嵐 (arashi - storm, tempest) for instance, or 潰 (tsubusu - to crush). 串 (kushi - skewer) and 貼 (haru - to stick) are both seen in streets everywhere as part of 串カツ (kushikatsu - Osaka-style meat skewers), and 貼り紙禁止 (harigami kinshi - "posters forbidden").
虎 (tora - tiger) and 熊 (kuma - bear) finally get their characters, though we may need to put them in a 籠 (kago - cage) and lock the 鍵 (kagi - key). We can now eat 鍋 (nabe - pot or stew) and 麺 (men - noodles) without resorting to hiragana. 狙 (nerau - aim), 袖 (sode - sleeve), 誰 (dare - who?) and 虹 (niji - rainbow) are other surprises.
Among place names we have 岡 (oka) as used in Fukuoka and Chizukoa, 奈 (na) for Nara and 阪 (saka) for Osaka.
Some other characters are still rare, overall, but show up in single words that are reasonably common. 鬱病 (utsubyou - depression) has become a poster-child for people opposed to this change; "utsu" is complicated and rare - even one of the components is very rare in itself - and really only used in this single word. Of course, people will mostly need to read it, not write it, and chances are this one will see more use too, once it becomes more familiar.
Overall, the change is a non-event for most people; they already know the characters, or will easily pick them up. Newspapers will adopt some of them and skip others for now. People will use them or not as they see fit. The real significance really is the fact of the increase in itself. It's a affirmation that kanji aren't going away, and an acknowledgement that technology has a role in driving language changes.
#1 You can use other characters if you add "furigana" - tiny kana characters indicating the pronunciation - to them. Book and magazine publishers sometimes use furigana for rare characters, and childrens and young adult books use them for any kanji the readers aren't expected to know yet. This is one reason that YA literature can be good practice material for adult language learners too.