I'm going through OWASP Cryptographic Storage Cheat Sheet and it says:
Encryption keys should be changed (or rotated) based on a number of different criteria: (...) After the key has been used to encrypt a specific amount of data. This would typically be 2^35 bytes (~34GB) for 64-bit keys and 2^68 bytes (~295 exabytes) for 128 bit keys.
Are those exact or minimum values when the key should be rotated? Also, why it should be rotated after large data volume is encrypted?
When you encrypt data securely using a block cipher, you use a mode like CBC or CTR with a MAC, or an AEAD mode like GCM or OCB. These modes all essentially randomize the blocks such that two separate input blocks with the same data don't encrypt to the same output block. Essentially, the behavior we want here is that the output stream is indistinguishable from a truly random of data.
However, when you encrypt more than 2^32 blocks (2^35 bytes) for a 64-bit block cipher, or 2^64 blocks (2^68 blocks) for a 128-bit block cipher, then we would expect to see a collision on the input to the block cipher, and therefore that an output block is repeated. However, in some modes, such as CTR, we'll never see a collision, and this violates indistinguishability, which may lead to attacks.
Note that piece of text you quoted is not correct: it's not the key length that matters, but the block size. For example, Blowfish can be used with many different sizes of keys, but it will always use a 64-bit block size. Similarly, AES will always use a 128-bit block size. When using a block cipher, we generally prefer 128-bit (or larger) block sizes these days, mostly for this reason.
This is a maximum: you should not encrypt more data than this, but you can of course encrypt less. If you need to encrypt many pieces of data with the same key, you can take your secret, generate a unique salt per message, and then use something like HKDF (e.g., with HMAC-SHA-256) to derive both a key and IV for each message. Then your key is technically different each time and you don't have to worry about this problem unless you have a single large message.
