My legs were surprisingly sore the day after my half marathon, a sure sign that I ran beyond (distance and pace) my preparation.
The best explanation on muscle soreness I've located can be found at Time-to-Run. As the article states, the muscle and connective tissue are damaged, and you should not attempt to run again until that soreness subsides.
That doesn't mean you should sit on the sidelines until the pain is gone. To the contrary, I recommend daily non-running related physical activity such as swimming or cycling to maintain cardiovascular fitness. Although you won't elevate heart rate, I think even walking is a lot better option than doing nothing (not in terms of speeding recovery but to prevent deconditioning).
Do anti-inflammatories help? According to one study using eccentric weight lifting to induce soreness, the level of pain and even Creatine Kinase (the marker for tissue damage) is reduced, however the ability of the muscle to perform did not recover any faster sportsinjurybulletin.com (the fact that ibuprofen appears to reduce CK levels quicker makes me think ibuprofen is not a bad option).
You can try some gentle work using The Stick, or Foam Roller, but the best advice may be to reduce the level of damage you suffer in the first place through training. Stuff like eccentric leg presses or downhill runnning may help some, but the bottom line is you probably need to do hard, long runs. In other words, races.
Think about those Marathon Manics who able to run back to back marathons every weekend and even two on a weekend. There is definitely something different about those folks. It's not that they are able to recover faster, but I believe they've been able to train themselves to the point that they experience minimal tissue damage. Running a marathon every week will eventually do that, at least compared to runners who race a marathon twice a year. But outside of that amazing group of folks, who is willing and able to do it?