Apply it to real world and you'll immediately see the benefits of map visibility. How well would we be equipped for war without radars? How valuable is stealth warfare etc?
Being able to sneak up on opponents is too strong of advantage to let any team have for free. Wards allow you to prepare for incoming ganks, to move quickly on map objectives, and even for some champions provides as a way for escapes.
The most obvious understanding of this comes to bottom lane. As a support, your job is to let you carry farm as much as humanly possible without being interrupted. Wards allow you to watch for incoming jungle ganks, roamers, etc.
The other role whose main job, among clearing neutral minion camps, is warding, would be the Jungler. As the bottom lane support is, quite frankly, stuck at bot lane, the roaming Jungler has a better chance of passing through areas that could use wards that would make or break a game. Until lane phase is over and the support starts to move around, it is important for the Jungler to ward the rivers to prevent ganks on top and mid, attempts on dragon and baron, counter jungling attempts by the enemy, as well as to provide constant vision of the other team so that your own knows when it is safe to pursue certain objectives.
For example, a ward in their jungle near the Blue Buff would show you that the enemy Jungler and/or Mid is currently not killing it and it is somewhat safe to steal it. At the same time, if the ward shows you both their Jungler and AP Carry are at the Blue Buff, it is safe to do a 3/4/5vs2/1 team fight on bottom since you know they will be receiving no reinforcements.
Seeing the map just allows you to do/access/defend/attack safer, smarter, more efficiently.
Best Answer
UPDATE 2016-06-23
Champion Shards can now be used to upgrade champion mastery levels.
To get mastery tokens for level 6 you need to earn an S- at the end of a game and to earn mastery tokens for level 7 you need to earn an S rank at the end of a game with a level 6 champion mastery.
To unlock level 6 combine 2 Mastery tokens with a champ shard of the same champ, OR the champ permanent of the same champ, OR 500 blue essence.
To unlock level 7 combine 3 Mastery tokens with a champ shard of the same champ, OR the champ permanent of the same champ, OR 600 blue essence.
How do you earn free loot?
You can earn free chests or earn free keys. To win free chests you, or some one you have queued up with must obtain a S-, S, or S+ rank. Only 4 chests can be obtained within a month and only one chest per S rank per champion per season. (Important to note if someone you queued up with gets an S rank and you earn a D rank on lets say graves, graves will count towards the only 1 per season).
To obtain keys, you must win a game and after winning you have a chance to obtain a key fragment. 3 key fragments combined equal one key. Then use the free key to unlock a chest.
What's the deal with essences and why are they so important?
There are two types of essences champion and cosmetic. Champion essence comes from disenchanting champion shards, which then can be used to unlock a champion permanently.
Cosmetic shards are for skins, champion and ward. You can earn more cosmetic shards by disenchanting skins. You can then use the cosmetic shards to unlock a skin permanently.
What do the different options mean when dealing with Redeeming Re-rolling Disenchanting and Upgrading?
Redeeming - lets you use the champion or skin for 7 days.
Re-rolling - requires three shards of the same type, guaranteed to give you a champion or skin that you do not already own.
Disenchanting - gives you more essences of that type.
Upgrading - use essences to make that shard a permanent unlock.
That should cover how it all works.