The reason why you have to shoot at those speeds are the ones Morgan mentionned.
I'd like to ad that it also has to do with the light frequency of a particular location.
The rule is that if you live in Europe, your speed will be 1/50 (wich coresponds to 50 Hz)
If you are in the States, it will be 1/60 (wich corresponds to 60 HZ).
If you shoot in Europe with 1/60 under artificial light (your house, the street...) you'll see a strong banding appear.
That's because your speed is not in alignement with the light frequency.
You have to think AVCHD like if it was a compressed JPEG. Let's say at 6 or 7 quality to simplify.
It's a "good" format for adquisition, but a really really bad format for editing.
The image recorded by your consummer camera is degradated quite a lot to be able to handle the amount of datas that motion involve.
You need to stop the degradation in post production, so you'd need to convert AVCHD to a suitable editing format.
AVCHD requires also in post a lot of calculations that tend to slowdown the editing.
Although many softwares can edit natively AVCHD, it's not recommended at all. So you need, to keep going with stills images, to save a Tiff from this Jpeg before you start to work on it.
The "tiff" would be your editing format. It could be Prores, DNxHD, HQX depending on the brand-system you're editing.
In the editing process, you'd prefer to work with low-res versions to speed-up the workflow and then on the color grading stage, you'd work with high quality versions to maintain intact the datas.
So you need to do an inquiery according to your system of how this can be done.
Then you do a master at the max quality possible.
From this master, you will create videos for the web, dvds, whatever your needs are.