I have a "templated" post for this which may help:
So you can give yourself costcondritis/tietze syndrome/etc. by aggravating your costal cartilages by doing this.
Since the origin of the pecs is on the sternum and partially on the costal cartilages, if you haven't used your chest in resistance trainig much before the connective tissues aren't as strong. Thus, when you do a deep movement where you erally stretch out the pecs like dips, the origin of the muscle on the costal cartilages can start to pull them away from the costal facets on the sternum
Obviously, this creates inflammation and in general can lead to some popping and hurting.
Best thing you can do is REST AND LET IT HEAL. Take your anti-inflammatories to help decrease inflammation and allow healign to occur. Fish oil is always good. Massage to teh area works as well.
Ice if it helps. Otherwise, once it starts feeling better use heat to stimulate more blood flow to encourage faster healing from the body since the area does not get a lot of blood flow as it is bone/cartilage in that area.
Keep the muscles working through full but non-weight bearing and non-painful range of motion so that the muscles don't tighten up on you, and the muscle movement will also help stimulate blood flow to the area as well.
Usually, if you didn't chronically aggravate you should be able to get back to exercising within a week.
Just avoid the exercise that aggravates it for a while, and focus on other exercises that are less intense of the chest... and then work your way back into these SLOWLY so you don't reaggravate your condition.