Why are collision attacks important when talking about MAC schemes?
The attack outlined by Drlecter is valid for any deterministic MACs (that is: with the MAC a function of message and key) with an iterated structure and an $n$-bit state. It relies on internal state collisions, expected to occur after about $2^{n/2}$ messages (the birthday bound), that can allow forgery once discovered. I'll illustrate this in the case of CBC-MAC with a 64-bit block cipher (e.g. 3DES) operating on fixed-size 3-block messages (24-byte) with 8-byte MAC (the full block cipher width).

The adversary asks for the MAC of $2^{33}$ messages with random first two blocks and the last block fixed. It is expected that a collision occurs, and the adversary then has $(M_0||M_1||M_2)$ and $(M'_0||M'_1||M_2)$ having the same MAC, with $M_0\ne M'_0$. So far, this is not a break. But notice that, with certainty, the collision has occurred after the second block, and continued after the third.

Now, for any last block $M'_2$, $(M_0||M_1||M'_2)$ and $(M'_0||M'_1||M'_2)$ collide with certainty. That allows the adversary to predict the MAC of one of these messages by querying for the MAC of the other, which is a break.

Thus a deterministic MAC with $n$ bit internal state (and output, but in itself truncating the output does not help much), after $q$ queries with the same key, has become unsafe with odds about $q^2/2^{n+1}$, for $1\ll q\ll2^{n/2}$. E.g. after $2^{60}$ queries, HMAC-SHA1 has odds $2^{-41}$ to be vulnerable.

Note: This attack (or any generic attack with similar result that I am aware of) does NOT in general apply to iterated MACs with an $n$-bit output. E.g. HMAC-MD5 or CBC-MAC-AES truncated to $64$-bit are not awfully vulnerable to this attack; after $2^{40}$ queries odds that the attack works are $2^{-49}$.