This is basically how it works:
A person will use a trojan to gain control of 10,000+ computers (not as hard as you would think).
They then export a program to those computers (also known as nodes or zombies) that makes repeated attempts to access a website. Causing the server or "website" to hang.
Think of it this way, each one of those 10,000 computers send 100 attempts.
That's 10,000 x 100 is one million(1,000,000).
Often it is much higher than 10,000 and much higher than 100 attempts.
The server gets blogged down and starts to "deny" service to legit customers trying to connect to that server aka "website".
That is DDoS in a nut shell.