Interesting. When I took modern foil fencing classes, it wasn't dangerous in the slightest. The training foils (the long thin bendy swords that people usually think of when they think of fencing) are extremely flexible and have a button on the tip. You score a point by pushing that button against your opponent in a score zone of their body. You wear an electrified vest over your torso with a long wire leading back to the score machine, and when the opponent's sword tip touches your vest it completes an electric circuit. If that circuit is completed at the same time the opponent's foil tip button is depressed, they score a point. Both people have full cage facemasks, and there's almost zero chance of injury short of tripping.

Epee fencing, on the other hand, uses a heavier, stiffer sword, so when you poke somebody it can bruise.

Sabre fencing is the third modern fencing style, and allows you to score points with the blade (slashing) instead of just the point as with foil and epee. I've never done it but have heard it is the most physical style of the three.

This photo is of an epee match, but displays the still considerable flexibility of the weapon and the electrification wires that trail behind each combatant. https://en.wikipedia.org/wiki/%C3%89p%C3%A9e#/media/File:Fin...