So now we've answered the question lets look at why this happens.
On the surface it seems ridiculous nonsense - how can two flights be cheaper than one?
Well, it's because with point to point air travel, more flights = inferior product so you are not paying per flight, you are paying for transport from A to B. Airlines have flexibility in how they offer this product - they can offer direct flights from A to B or they can use hub airports to get you from A to B via C.
In this case we have to look at Manchester to Miami and London to Miami as two different markets - different customers, different needs and different competition. There will be a small overlap of people who can go to either, but generally, people won't look for flights like that.
If we look at the dates the OP wishes to travel we can see that there is something of a price war going on between indirect carriers. There is loads of choice.
You can fly Finnair via Helsinki, Lufthansa via Frankfurt, Austrian via Vienna, American via Philadelphia or KLM via Amsterdam. They are all similar money. So if you pop into the market here and try to flog people British Airways via London for £600 you'll sell no seats. So, to remain competitive, you need to price at a similar amount to the competition. Which is what has happened and is how we arrive at that price for Manchester to Miami via London.
Lets move to London.
The same indirect options exist but are around £100 more expensive. So, you've no need to go in so cheap. Plus, the British Airways product is now a demonstrably superior product - it is a direct, non-stop flight which is considerably quicker, so people are willing to pay more for it. So if you price it at the same price as the Manchester flight - or perhaps even less because you don't need to sell the shorthaul seat - you will very quickly sell all available seats for far less than you could reasonable have expected to get for it. So you don't.
And thats why its cheaper to fly Manchester to London to Miami than it is London to Miami on the same plane.
So why can't you just buy the Manchester ticket and get on at Heathrow? Because airlines are not stupid - it's therefore in the terms that if you no-show for any segment on a flight booking, the rest of the booking is cancelled.
btw - those prices are insanely good. Book them ASAP. It doesn't get much cheaper than that to cross the Atlantic.