Clear[p, x, n];
yi = {1.2, 2.3, 3.6, 4.9, 5.9};
xi = {1, 2, 3, 4, 5};
data = Transpose[{xi, yi}];
nlm = NonlinearModelFit[data, p x^n, {p, n}, x]
p1 = ListPlot[data, PlotStyle -> Red];
p2 = Plot[Normal[nlm], {x, 0, 6}, PlotTheme -> "Detailed",
              FrameLabel -> {{"y(t)", None}, {"x", "non-linear Fit"}}];
Show[p2, p1]

Transpose in this case essentially combines the two lists (like using Partition[Riffle[xi, yi], 2]). Nasser made a 2 x 5 matrix (each list as a row) and the transposed it to get the pairs of coordinates as a 5 x 2 matrix. This gets you the XY pairs. You can see this if you run the steps below:

Clear[xi,yi];
yi = {1.2, 2.3, 3.6, 4.9, 5.9};
xi = {1, 2, 3, 4, 5};
{xi, yi} // MatrixForm
Transpose[{xi, yi}] // MatrixForm

And how would you find the residuals?

Options are build-in. Help for more information:

yi = {1.2, 2.3, 3.6, 4.9, 5.9};
xi = {1, 2, 3, 4, 5};
data = Transpose[{xi, yi}];
nlm = NonlinearModelFit[data, p x^n, {p, n}, x]
nlm["FitResiduals"]
nlm["StandardizedResiduals"]
nlm["StudentizedResiduals"]
nlm["EstimatedVariance"]