A simple way to explain it, but a little more technical:
1. The wallet software/hardware gather entropy from random sources. This entropy is 128 bits long.
2. The entropy gets hashed and the first 4 digits of the hash are appended to the entropy. Now the bits you have are 128 + 4 = 132.
3. Then the sequence is split in 12 pieces of 11 bits each.
4. Each piece is converted to decimal and you get 12 numbers that correspond to words from the BIP39 wordlist:
https://github.com/bitcoin/bips/blob/master/bip-0039/english.txt5. The mnemonic phrase is converted to a seed using PBKDF2 function (it's a hashing function actually). The seed is 512 bit long.
Now, the private key is generated from this seed (the 512 bits).
So, if you know the original entropy (step 1), you can derive the seed (step 5) that produces the key.
But, technically speaking, the mnemonic words and the private key are not the same thing.
The mnemonic is nothing more than the representation of the original entropy. Is it important? Yes! Too much! It can help you recover your wallet.
The private key is also very important because it does all the work. But, you don't have to back it up, because it's much easier to backup 12 words, since you know that they will produce the same key all over again.