The two "flat" coils define planes. There is an axis that is perpendicular to the planes defined by the coils. The axis goes through the center of both coils. The coils are a distance x apart.
Let k be the unit vector that points along the axis through the center of the the coils.
Then the vector from the center of the large coil to the center of the small coil is x k. Sorry to use non-standard notation but the distance between the coils specified by the problem statement is x. (It would have better to be able to use z!)
Let a be the vector from the center of the large coil to the element dl. We have already defined r as the vector from dl to the field point at the center of the small coil. Therefore
x k = a + r
So r = -a + x k
a is just the vector from the center of the large coil to a point on its circumference, so we can write as R1(i cosθ + j sinθ).
Therefore r = -R1(i cosθ + j sinθ) + x k.
Each dl has a different r, but the magnitude r is the same. It does not depend on θ.
Doing the cross product
dl x r = [x(j sinθ + i cosθ) + k R1]R1dθ
You have three terms to integrate from 0 to 2π, but the terms containing sinθ and cosθ will integrate to zero; therefore, the only term that will contribute is from k R1^2dθ
I hope this helps.