After writing this answer, I noticed there are a couple alternative explanations that might be interesting to mention, so I'll add them as well.
Explanation 1
What makes something into a black hole isn't exactly how much mass it has, but also how compactified it is. In principle, any amount of mass can form a black hole, as long as you compactify it enough.
The size needed for some amount of mass to form a black hole is know as the Schwarzschild radius. Roughly speaking, if you pick an amount of mass and manage to compress it below the Schwarzschild radius, you'll have a black hole. It is given by a simple expression. Namely,
$$R_S = \frac{2 G M}{c^2},$$
where $M$ is the mass, $c$ is the speed of light, and $G$ is Newton's gravitational constant (which sort of measures how intense gravity is). For example, for something with the mass of the Earth, the Schwarzschild radius is roughly $0.88$ cm, while for the Sun it is about $2.9$ km (I must admit I didn't double check the computation, I'm trusting Google on these numbers, but they are pretty much what I remember).
Hence, the black hole stays a black hole while it evaporates because it is shrinking while it is losing mass, and always shrinking enough so that it is always at the correct size.
Explanation 2
The second way of thinking is a bit less familiar. It turns out that black holes aren't really objects, but rather regions in spacetime. In fact, this is so true that black holes are what we call vacuum solutions: there isn't matter anywhere in the spacetime. All of the mass of the black hole is there due to effects of gravity itself. Another way of thinking it is that a black hole is so collapsed that its mass is entirely due to gravitational energy.
It is a bit harder to grasp this concept, but once you get it, the rest is simpler. The black hole stays there because it isn't "made" of anything. There isn't a star just below the event horizon waiting to come out. There is nothing there, but gravity. As it loses mass, gravity weakens and it gets smaller, but there isn't anything behind the horizon to come out.
Edit: the question "What do we mean when we say that black holes aren't made of anything?" later asked for a more technical discussion of parts of this explanation. I suggest checking it out.
Explanation 3
The third explanation might be a bit simpler than the second. Once something falls into a black hole, that's it. It can't come out. Ever. By the very definition of what a black hole is. Hence, as the hole shrinks, there is no way something could come out of the hole to be its "body". That would violate the very meaning of what is a black hole.
This is a simplified answer. Since OP doesn't have formal education in Physics, I might have overlooked a few details and nuances in here, but I did my best to keep the answer as faithful as possible.