Basically, what you really care about is "How likely is it that any given molecule of this stuff will randomly break down at any time?" because if you know that and you are working with billions and billions of molecules, that gives you a very good idea how long the drug will last overall.

But even drugs with short shelf-lives are stable enough to make that probably very low, so instead of saying "In any given second there is a 0.000000000000000000000000000000001 percent chance of one molecule breaking down." we measure it indirectly instead. And that's what a half-life is: it takes the probability of each molecule breaking down and is an estimate of how long it will take for there to only be half of the substance left.

Let's use a model as an example: let's say you have a school of 1000 children, and you give them each a ten-sided die. You have them all roll the die at the same time, and every child who rolls a 10 gets eliminated.

In the first round, about 100 students will be eliminated, leaving 900. Next about 90 are eliminated, leaving 810. This keeps going with the number eliminated getting lower or lower until eventually there are only one or two students left and it becomes very hard to predict how long it will take to eliminate the last few people. The number or rounds necessary to get to about 500 students would be the half-life in this scenario.