Even if scientists found a way to shrink a human down tot he size of an ant, there would be too many problems with the process for the human to survive. But, this is a movie, so we can disregard any rational thinking and assume everything works out just fine when humans are shrunk to 1/1000th of their original size.
The movie actually does a good job explaining one concept of shrinkage; and that is increased density of the individual after being shrunk. When a person is shrunk down, the space between their atoms is shortened, meaning they still have the same amount of mass with the same number of atoms, but have a drastically reduced volume for those atoms to reside in. The average density of a human being is 985 kilograms per cubic meter. If shrunk to 1/1000th of their original volume and kept their same mass, their density would become 1000 times denser resulting in the density of the shrunken individual to be 985000 kilograms per cubic meter. This would result in a shrunken human, less than half an inch tall, to have all the mass of their normal size, causing a massive density shift.
The movie explains how when Ant-Man, in his shrunken state, punches someone, in carries all the weight of the punch, over a smaller surface area, equal to "being shot by a bullet." And while the density shift in the previous paragraph does confirm the effect itself, it most likely would not have enough power as a speeding bullet, maybe a low caliber bullet. And even with this new-found "strength", Newtons 3rd law still applies. If Ant-Man were to punch someone in his shrunken state, his now shrunken figure would feel the same recoil a gun would from shooting said bullet. And being as small as Ant-Man is, would most likely kill him as well as the bad guy he punched.
Below is a video explaining these two concept and even more problems with shrinking humans down to that size. Guest star Ant-Man (Paul Rudd) himself will help explain the problems with his own movie. It is a tad longer than other videos I found, but well worth the time.