You can partly get to grips with exponential increase by thinking about what is called The Malthusian Catastrophe.
Thomas Malthus claimed that the world's population is growing geometrically; though food production is only growing arithmetically.
What did he mean? In terms of food production, he was talking about the rate of new acres open to agriculture each year. He believed that the rate was fixed. Thus the food supply works, for example, like this: 100, 102, 1004, 1006.... 1012.... etc. That is, the acreage grew by two acres per year in Malthus's' day. It was a fixed rate of increase which never changed.
On the other hand, populations don't work that way. Basically, the more adults who can have children, the more babies there will be. However, that rate isn't fixed as in 2, 4, 6, 8... etc. This is shown, or at least it was, in terms of the entire population of the world. Instead of 2, 4, 6, 8... etc., it was more like : 1, 2, 4, 8, 16, 32... Instead of the simple addition of 2, we have a number being doubled every time. Thus if you compare 2, 4, 6, 8, 10 with 2, 4, 8, 16, 32, we have a difference of 22 even though both progressions included only four changes. The geometrical increase ends in 10, whereas the geometrical increase in 32. Both rates included only four changes or progressions. Clearly, as the progressions increase, the gap between the arithmetical ratio and the geometrical ration will keep on widening. Or, in Malthus’s case, the population increase will keep on outstripping the increase in food production – resulting in starvation, etc.
Another word for geometrical increase is exponential increase. Exponential increase is applicable to most or all living organisms. It's also applicable to human populations. Another way of putting this is to say that exponential increase is proportion to the given number being increased. With arithmetical increase, it is just the addition of 2 (or 3, 7, etc.) each time. That isn't the case with exponential increase. The larger the number, the larger the increase.
Many arguments against Malthus’s argument have been advanced. That is, even if populations increase, it isn't a necessary, or mathematical, fact that more people are likely to starve. However, even if Malthus’s prophesies were false, which they were (in the UK at least), the exponential increase of families is still (largely) true. Or at least it's true given other (many other) conditions, such as: that all off-spring themselves have families and that what happened to the first family, will happen to all further families generated by that first family. In other words, all the women need to be fertile, all the children need to survive into adulthood, and all the children need to get married, etc. If all these factors occur, then there will indeed be an exponential increase in the population number.
We can say that if all the conditions remained the same for Malthus's arithmetically increased food production, he would have been right.... But that's just it – he didn't foresee the possible other conditions in either the geometrical or the arithmetical case.
However, you can have exponential increase that isn't precise or which fluctuates. (Does that automatically stop it from being exponential increase?) That is, in theory, a couple could have six kids. Those six kids may get married and each have six kids. That would amount to 36 people in two generations. And if those 36 people did the same thing, that is, each have six kids, then the number of people would now be 36 times 6, which is 216 people. So from one couple, and in three generations, 216 people have been produced! That's an increase from 2 to 216 in twenty or so years. Now that's just one family that has produced 216 people in just twenty or thirty years. What about ten or a hundred families?
Take 100 families which have each produced 216 persons in twenty years. That's 216 times 100, which is 21600 people in twenty years. That is, 100 families have produced 21600 people in twenty years.