Look at it that way:
You move through x steps, each step takes y time, so the total move time is x*y
So you move once in 1s (x=1, y=1). You moved during a total of 1s.
Next, you move twice, each time during a half-second (x=2, y=1/2). You moved during a total of 2*(1/2) = 1s
Next, you move four times, each time during a quarter-second (x=4, y=1/4). You moved during a total of 4*(1/4) = 1s
We can continue like that as much as we want. If we keep x=n and y=1/n, then the end result will always be n*(1/n) = 1
As you can see, the result is always 1 and is actually independent from n.
Now what happens if n=+∞ ?
The result is independent of n, so it will still be 1.
We can talk about it as a function, with f(n) = n*(1/n). That's the same thing that we did above, with n=1, n=2, n+4 and n=+∞ respectively.
The result at +∞ is called a limit. it is where the function tends toward when we get closer to infinity. This case is pretty simple as it is always equal to 1, so of course it tends toward 1. But for example, 1/x tends toward 0, as the bigger x, the closer to zero is 1/x
Limits are what Zeno didn't have, hence the (apparent) paradox. Without limit, you would try to calculate +∞* (1/+∞) but that would be +∞/0, which is undetermined. The trick with limits is that you can rewrite the same function in different manners (as did here with n* (1/n) = 1) where you won't actually have an undetermined.
This is of course not always possible for all functions, but in this particular case, it is.
Limits were discovered only a few centuries ago, and opened whole new fields in mathematics.
To start with, it enabled movement. A little known fact is that nothing could move before the nineteen century. Which was a good thing, as gravity didn't exist before Newton, everything would have just floated away.(*)
(*) may not be entirely exact
In a more intuitive manner, as your number of steps rises, the size of each step drops. When you tends toward an infinite number of steps, each step tends toward zero. So you have an infinite number multiplied by a size of zero.
Every finite number (bar zero) times infinity equals to infinity. After all, if you have an infinity of something, you have any quantity of the something an infinity of times.
Every finite number time zero equals zero. After all, if you have zero something, you have zero anything.
So what happens when you have an infinity of zero, or put another way, zero infinity? Well, it could literally be anything. It depends on what zero and what infinity, if you want.
In this particular case, it happens to be 1.