First, I will confirm that your except is correct.
The meta currently is a mage mid, bruiser top, jungler in the jungle,
and ranged carry and support bot.
- Mage/Caster - Mid
- Bruiser - Top
- Ranged Ad - Bot
- Support - Bot
Next, let us look at why the current meta works/makes sense.
We need to understand each of the roles defined above to understand what they do and how they fit into the current game state. The LoL Wikia's List of Champions shows the designation each champion received from Riot as it pertains to their role on the team. They include a few types that aren't in the list above, such as Assassin which includes champions that can be Bruisers, Casters/Mages and/or Junglers. Note that some champions can fill multiple roles (just not usually in the span of one game). Rather than explain every champion and how they can be played, I will assume that you have done some research yourself and have a general understanding of how any individual champion works. Back to answering your question!
One thing that should jump out at you when you look at the list of positions (top/mid/bot) is their place on the map and what objectives that puts them near. We will start, as with most things, at the Top.
Top lane is, by far, the most isolated lane. The early game objective (aside from jungle buffs) is dragon. Top lane can't really participate in the fight for dragon without abandoning the lane, seeing as the dragon camp is between Mid and Bot lanes. Some players will use the Teleport summoner spell to have more of a map presence when they find themselves in the Top lane but this is at the expense of another summoner spell that could prove useful against your lane opponent. Being this far from dragon also means that if someone else on your team (Jungle or Mid usually) comes to help you against your lane opponent, you are no longer pressuring dragon. Sometimes showing the enemy team that your Jungler is top is an open invitation to the enemy mid, jungle and bot players to go for dragon and take it or force a 4v3 in their favor. So Top, being unable to participate in the fight for dragon and not wanting to pull other allies into the Top lane, thus weakening you map position relative to dragon, needs to be resilient (but they should also probably be able to do some damage to try to get the enemy team to help their Top laner, thus strengthening your map position relative to dragon). This usually means that champions with the ability to take a few punches and dish out some damage (hence the nickname Bruiser) end up Top.
Working our way down from the Top we come now to Mid. Champions who end up in Mid tend to be, as you pointed out, Mages/Casters. The difference between the two, for purposes of this answer anyway, is this. Mages build ability power and tend to derive their strength from their abilities rather than their auto-attacks. Champions refered to as "Casters" tend to include champions like Talon and Pantheon. Casters derive a lot of their strength from their abilities but do it by building attack damage and have some use for auto-attacks. The similarity between these two designations is that they both get their strength/offensive capability/damage from the abilities. There are two ways to make a champion's ability stronger (1) level it up (2) buy items with stat that the ability uses in its damage calculation. Both of these methods can be achieved by being the only champion in a lane (rather than being paired with an ally). You get all the experience in each minion wave and you get as much of the gold as you can successfully farm (which then lets you build items). Seems pretty simple, and it is. Mages/Casters then tend to be able to dish out a lot of damage (but unlike Bruisers are squishier) and can therefore participate in the fight for Dragon. Their squishyness also means they might need more help from their Jungle but unlike Top lane, this doesn't take away from pressure at dragon. In fact, if you can kill their mid you may be able to initiate a fight at dragon 4v3 in your favor.
Before we get to Bot, we will look at why we have a Jungler on the team. The Jungler takes advantage of the experience and gold and provides map pressure and a variable degree of risk to the enemy team (if you can't see the Jungler they could be anywhere). The Jungler is usually a champion that can hold their own against the jungle monsters and gank enemy champions and therefore usually have some form of crowd control. The Jungler usually has Smite which can be very useful in securing dragon, jungle buffs and Baron and can place wards to assist in keeping track of enemy champions movements.
That brings us to Bot. There are usually two champions down in Bot lane including the attack damage carry (ADC) and the support. ADCs tend to get most of their damage from their Auto-attacks augmented by an attack steroid available to them through their kit Tristana's Rapid Fire, Graves's Quickdraw, Kog'Maw's Caustic Spittle and Bio-Arcane Barrage are a few examples of attack steroids. This means they want items and to be able to buy items they need gold. To get the gold they needs minion/champion kills but unlike mages and bruisers they are both squishy and lack strong abilities that would deter opponents who are trying to kill them. This is where Support comes in. Support, simply put, supports the ADC. This can mean sustaining the ADC with heals, like Soraka, setting up kills like Blitzcrank or some combination of both (see Alistar).
Mage/Caster - Mid
Derives damage from abilities, wants more experience to level up abilities.
Bruiser - Top
Needs to not die but also needs to be a threat.
Puts pressure on lanes and secures objectives.
Ranged Ad - Bot
- Support - Bot
Get the ADC online and doing as much damage as possible.
This of course all breaks down after the Laning Phase ends but I hope it helped!