Is it possible to know why an airline operates specific equipment on certain routes? I've spent far too long staring at flight tracking sites, and occasionally I think I'll notice some pattern in what types of aircraft a given airline uses for certain scenarios/routes. 

For example, Delta seems to operate a lot of MD-88's on short-haul routes that other airlines would likely use ERJs or CRJs on. They also seem to use 757s on quite short hops (ATL to southeastern cities ~1hr away).

Another example would be why British Airways operates 747s to some cities, and 777s to others, and exactly how they decide where to use which aircraft.

Is there a good reference for figuring out the reasoning behind why airlines use a given aircraft for a given route? Or are there too many different factors and too much corporate secrecy to really know for sure?
There are many factors, but you can often predict what class of plane will be used for a particular route by the distance and demand.

For smaller cities, they want to fill the planes, but they also need several flights per day to cover the fixed costs of serving that airport (and make things convenient for customers), so it's often better to have a commuter jet bouncing to the nearest hub and back several times per day day than one large plane that carries an entire day's worth of passengers in one shot.  For larger non-hub cities, they'll do roughly the same thing but with mid-size planes.

Between hubs, they'll run the largest narrow-bodies to connect all the various small cities not served by a common hubs, plus hubs themselves tend to be large cities with plenty of native demand too.

For international, they'll run wide-bodies because they need higher fuel capacity, and there's rarely a convenient time for customers due to the long flight times combined with crossing several time zones, so you might as well pack them into as few planes as possible.

Notably, some carriers fly several different types, each optimized for specific route types, while others fly just a few types (or even just one) because that makes maintenance, pilot training and fleet scheduling easier--but may limit what kinds of routes they can fly profitably.  When airlines merge, they'll usually have similar fleets--and in cases where they don't, they'll quickly start dumping types they don't like and replacing them with types they do.