Why use CCM instead CBC-then-MAC? Is there any reason why CCM was created with encryption/decryption being done with a counter versus using CBC-then-MAC?
CTR mode was previously considered a relatively brittle scheme, and in a sense it is. Repeating the nonce will repeat the counter, and repeating the counter will repeat the key stream. This means that the security of the plaintext is broken rather spectacularly on nonce reuse.

However, using CTR has many advantages over CBC:


You can use a nonce instead of a randomized IV (deterministic vs random);
You don't need to bother with padding or ciphertext stealing (no expansion of ciphertext above plaintext size);
Because of above it is not vulnerable against padding oracle attacks (decryption before verification is not vulnerable);
You only need to use the block cipher in one direction (simplification of the encryption scheme, less hardware required);
Online encryption/decryption is perfectly possible (immediate encryption / decryption without caching);
Ciphertext streams may be cached for efficient, byte oriented encryption / decryption (low latency);
As the counter values can be generated in advance, pipelined AES implementations may offer (much) higher performance - see the comment from Yehuda for a description (speed).


So nowadays about every AEAD cipher uses CTR mode underneath. And that's fine, as long as you do indeed provide it with a nonce.

The fact that you can also search within the ciphertext (skip to a specific offset) is of little interest for AEAD ciphers as you would generally not use the data without first verifying the authentication tag. So although it is an advantage of CTR mode, it doesn't make much of a difference within an AEAD cipher such as CCM.

One problem with CTR is that the construction of the counter itself isn't well specified (location of the IV, additional data, etc.). But within an AEAD cipher this is not an issue as the AEAD algorithm itself will specify the precise construction of the counter.



The question is more why the archaic CBC mode is still used that much. It's not as brittle as CTR mode when it comes to the IV, but that comes at a very high price, including additional requirements for that same IV.



Also note that the use of a CBC mode together with a CBC based MAC may not be such a good idea if the same key is used for both; it is easy to introduce conflicts between the two, threatening the security of a scheme providing both confidentiality and integrity of messages.