Some fighters use their physical weapons in the cage, and some people get into mental warfare. If you have the physical tools and on top of that start getting into mind games, that can be a highly deadly combination. Three guys have been most successful with this tool of mind warfar:
1. Anderson Silva
Putting his hands down, letting you hit him, dodging out of the way, and a variety of different antics have been used by Anderson Silva to throw off his opponent's games. He is able to prey upon fighters' insecurities and bring them out in the UFC Octagon. It didn't end up working against Chris Weidman, but it is important to note that if Weidman didn't land the fourth strike of the combination he fired at Silva to end the fight, the bout could have ended drastically differently. Because Weidman was getting angry and charged at Silva, and if he missed four straight shots after that, then that could have turned the momentum totally in Silva's favor; he was keeping the fight on the feet and starting to shift the bout into his control.
2. Nick Diaz
"Why is your nose bleeding?"
"Oh, we're throwing spinning s**t now?"
These are the sorts of verbal tongue lashings that Nick Diaz doled out to Carlos Condit in addition to slapping him in the face. Kudos to Carlos Condit for staying composed and not biting on the mind games, but many others aren't as mentally disciplined as Condit. Diaz's mind games in addition to his punching skills and endless endurance have made him a force at 170 his whole career. Go back to early in his UFC career when he fought Robbie Lawler; he was tagging Lawler and talking smack to him, slapping him, and Lawler got very angry and came at Diaz rushing in, only to be met with a straight punch that dropped him and ended the fight. Diaz is a master of boxing, cardio, and mental warfare.
3. Frank Shamrock
Frank Shamrock was tremendously strong in his mind, body, and spirit. He was another fighter who was able to expose his opponents' insecurities inside the cage. He would ask his opponents why they are breathing heavy, make a gesture that he is going to put them to sleep, and continually badger them with words and antics. It worked well for him in most of his fights and he used that mind trickery to mess with the head of the his foes in combat.
One thing to note is that using these techniques can be very effective, but if someone uses these techniques on you, you combat them and use them your advantage. Chris Weidman took the approach that if Anderson Silva stands there letting him punch him in the face, that's great, because now he's winning the fight. He's the one scoring points. Carlos Condit also was effective and stuck to his gameplan better than anyone against Nick Diaz on route to victory and showed incredible focus and discipline in the process.
You can change the way you perceive things. You can go into competition already thinking that if someone uses antics, this shows that they are not confident in themselves. Your opponent is no longer using his physical tools, but has to resort to other tactics. You can take the viewpoint that he is mentally breaking, running out of ideas, and grasping at straws, because he is insecure about using just his physical skills alone to win. That changes the way you frame things in you mind and the antics have a different energy because of your perception. Instead of a sign of confidence and bravado, now in your mind it is a sign of weakness in your opponent that he's resorting to these tactics; now you have the mental edge in your brain; which is the only point of view which matters in competition.
Additionally, you can mentally prepare to use your opponent's smack-talking and antics as an "anchor". So you can visualize your opponent doing what he normally does in terms of mental games, and the second he does that, you can visualize yourself successfully landing techniques on him. Basically conditioning your mind to get into a state where you're totally confident and in the zone when you see your opponent doing these things. For example, when you mentally rehearse reacting to something your opponent does or says, think of a time where you were totally on fire and in the zone and landing your techniques with total precision, accuracy, and confidence. Feel the feelings associated with this, hear the sounds, and get a clear mental picture of it all. The more you practice visualizing this success after picturing your opponent engaging in mental ware, then when it happens in combat, you are prepared for it and you have mentally conditioned yourself to get into a peak performance mindset when it happens. So instead of being a negative factor during competition, it becomes a positive one.