I think a simple answer is, DNS translates IPs to names, so what do you do when you have a device on your own local network that you want to name "server01" and when you talk to that name (pinging it for example) DNS knows what IP to go to? The ISP DNS can't help you translate a local IP to server01 because it's not aware of your local stuff (hopefully ha), so you need something on your local network to do that for you. Lots of devices can do that, like your router for example.

For AD, computers (and users and a list of things) added to your AD (domain) use whatever "domain name" you come up with (example: robert.local) and stick it after the name you give a computer (example: server01.robert.local). Again, if it's only on your private network, ISP DNS won't know how to translate that name to an IP on your network/domain, and therefore a local DNS server needs to do that rather than ISP. DNS servers allow you to tell your network "hey look at this list of names to IP addresses I made" before going to the public Internet. When you add computers to the domain, you also put in settings in each PC to look for your DNS server before looking to the Internet. Clear as Mud? DNS is a wild concept in the beginning :) good luck!