Why not ask them? Just send them an email saying that the offer sounds fair and you'd be happy to agree to it (assuming that's the case), but you're intrigued as to why they increased it by so much.

If I had to guess, it sounds like perhaps they had no idea how much the market rate was and then, after you countered, sought advice on this. Having learnt what the going rate is, they then offered what they thought was a fair price. Whilst it seems crazy to offer someone more than what they were asking, they might do so if they're planning to hire you full time.