Capacitors are made up of two plates (& so the symbol) which you can imagine as 'boxes' of opposite polarity and are seperated with insulation. 

As capacitors charge, the negative box keeps filling with electrons while the other (positive) box loses any electrons still in them due to repulsion from this negative box. Since the negative box is relatively empty to start with, electrons fill in very quickly. As their numbers increase, the capacity of the box reduces and the electrons repel any new electrons coming in, which slows the flow of electrons. 

Similarly for capacitor discharging, the now filled negative box easily looses its electrons to the empty positive box very quickly. But as their numbers start to even out, the flow slows down. Hence, the graphs potray an exponential relationship for capacitors when charging and discharging takes place. 

This is quite similar to you filling/emptying a balloon with water; it starts to fill/empty very suddenly at the beginning but slows down over a period of time. .