Question

how is the integral of velocity = position?

Answer 1

integral of velocity (with respect to time) is displacement. Consider a very small displacement:
\[\Delta \vec r=\vec v(t) \Delta t\]
because velocity times a very small increment of time will give you a small displacement. An integral just adds up all these small displacement elements and takes the limit as time approaches zero. So,\[\Delta \vec r =\int\limits_{}^{}\vec v(t) dt\]

Answer 2

If you don't understand the usual explanation, try looking it up in Calculus by Gilbert Strang. It contains the clearest and definitely the most insightful explanation of the integral and especially of Newton's formula I've ever seen.