AES-CBC only insures confidentiality, not integrity or authenticity.
If you look at the image down below, you can see how the Cipher Block Chaining Mode works for AES:
Source: Wikimedia Commons
For now, let's ignore all the other blocks of data, padding, etc., and just focus on the first block of data. The ciphertext is decrypted with the key, and then XOR'd with the IV.
Since the attacker can freely modify the IV, the changes to the IV are directly, bit-for-bit, reflected in the decrypted text. This property is called Malleability. In fact, all the blocks are malleable, since you can modify any block of choice and it'll be XOR'd into the result of the next block. Yes, you will "lose" one block, but depending on how the data is structured, that may be okay.
Isn't this really dangerous?
Yes, which is why AES-CBC must be used in combination with a primitive that guarantees integrity and authenticity. That means, if either the IV or the Ciphertext are modified, the decryption will fail.
Or, you can use Authenticated Encryption, such as AES-GCM, which will do all of this for you.