No expert, but have lots of experience personally and from helping others. The reason why you’ll see replies all over the place with this topic is shin splints is a general term that can refer to different things (medically, shin splints is an outdated term). Also, there are lots of different causes to the various maladies commonly termed shin splints.
Pain at front of shin, usually Anterior Compartment Syndrome or Stress Fracture. If the pain is pinpoint in a spot, more likely Stress Fracture, if pain is more diffuse, probably Anterior Compartment Syndrome (muscles swell up inside the inflexible fascia surrounding it, causing pain).
Pain on lateral side of shin, usually Lateral Compartment Syndrome or Stress Fracture.
Pain medially on shin, usually the lower half of the tibia, usually Medial Tibial Stress Syndrome (MTSS) – probably what you have. This in my experience is the most commonly seen version of shin splints. Medial Tibial Stress Syndrome is caused by the anterior muscles of the lower leg pulling on the bone (tibia), which irritates the outer lining of the bone (periosteum), causing periostitis (inflammation and pain). This can be caused by a variety of factors, many of which were hit on by the various replies, most common factor is too much too soon (improperly ramping up the increase in running volume). Also, running on harder surfaces (concrete is harder than asphalt) or poor running shoe selection are common culprits. Your arch problem may be problematic, as I’ve heard from people that proper arch support for people that need it can help a lot with shin splints.
What you can do about it: long-term fixes include strengthening the tibialis anterior muscle (dorsiflexion movements, many of which were shown in the video). Also, changes in running style is another long-term fix, such as running in Vibram Five Fingers. These shoes force you to run with more a forefoot strike vs. a heel strike, which can cause a lot of soreness and tightness in the calves initially (it takes some getting used to, so if you go this route, google the different break-in programs available). The proponents of this running style espouse that it is the way nature intended us to run, and it is great for long-term foot, ankle, shin, knee, and leg health when running. It is not the be-all and end-all of the discussion, as the heel strike style of running has been used successfully by runners for a long time. These are long-term fixes as they will not help with your problem right now, but will help prevent it from happening again in the future.
Short term fixes include RICE (Rest, Ice, Compression, Elevation). If you lay off the running for a few weeks, the MTSS will go away, but you have to slowly ease up the running volume after the rest period. If you have to keep running, there are things you can do to help. Compression is usually key, you need to get a good amount of compression without compromising circulation to your foot. An athletic trainer (if one is available) can help immensely with a proper tape job. If you have to go the do-it-yourself route, try a compression wrap that goes from the bottom of the shin, just above the malleolus (the bony protrusion at the ankle) all the way to at least the middle of your shin. What you are doing is compressing the muscles to reduce the stress or pulling on the periosteum. Make sure to ice afterwards, as the shins will still hurt (but less with good compression) and they will definitely hurt afterwards when you take off the wrap.
Also, I’d recommend you follow the advice others gave of going to a good running store and getting properly fitted for good running shoes. If you really tend to overpronate (feet roll inward) then you probably need some good arch support, either from a shoe or an orthotic, depending on the severity of your problem.