Eventually, Achilles catches up with the slower tortoise. His path will then necessarily include infinitely many (gradually decreasing) consecutive segments. Therefore, he would have to cover an infinite number of distances before he finally reaches the tortoise. But this is impossible, as no one can ever complete an infinite sequence of actions.
The seeming contradiction arises from the unspoken assumption that Achilles must become conscious of each individual segment during his motion. Intuition tells us that humans can only have a finite number of discrete conscious experiences behind them at any point in time. How is then Achilles' motion possible? Well, being conscious of the whole (path) does not imply being conscious of its (infinitely many) parts.
One can eat a bowl of rice without becoming conscious of every single grain of it.