Perseverance's umbilical data cable and bridles were cut by exploding bolts. This is the same method that Curiosity used.
Exploding bolts are very commonly used in space missions. This is for several reasons. First, the fasteners must be strong enough to hold the pieces together throughout all of the vibrations and motion a spacecraft must endure as it launches. There must be absolutely no chance of them separating, otherwise vital components could break. However, they must also be able to release at precisely the right time, and very quickly at that.
An exploding bolt is extremely simple, and has a very low chance of failure. Since NASA has been using these for decades, they know how to use them very well. And NASA likes using technology they already know how to use. Could they have made a fancy mechanism to properly disconnect the cables? Probably. But the more complicated the mechanism, the more likely it is to fail. And it's not like they are going to use those cables later.
This video does a very good job of explaining how explosive bolts (Hold Down Release Mechanisms) work on spacecraft (though it's not the one I remember seeing).
Edit: From the comments and other answers, I see that the communications cable was cut with an explosive guillotine. I'm going to leave this answer up, though, as it provides insight into how the other cables were cut.