In the wikipedia link that you give, they describe some ways of computing these intersection numbers in practice. The one I'm going to use is described in this paragraph:
One realization of intersection multiplicity is through the dimension of a certain quotient space of the power series ring $K[[ x,y]]$. By making a change of variables if necessary, we may assume that the point $p$ is $(0,0)$. Let $P(x, y)$ and $Q(x, y)$ be the polynomials defining the algebraic curves we are interested in. If the original equations are given in homogeneous form, these can be obtained by setting $z = 1$. Let $I = (P, Q)$ denote the ideal of $K[[x,y]]$ generated by $P$ and $Q$. The intersection multiplicity is the dimension of $K[[x, y]]/I$ as a vector space over $K$.
Step 1: Since your equations are given in homogeneous form, we start by picking an affine coordinate patch to work in. I will assume your point $P$ lies in the affine coordinate patch where $Z\neq 0$. In this patch your curve and tangent plane are defined by the polynomials $$F(x,y) = x^{p+1} + y^{p+1} + 1,\,\,\,\,\,\,\,\,\,\,T(x,y) = x_0^px + y_0^py + 1,$$ respectively, where your point $P$ is $(x_0,y_0)\in \mathbb{A}^2$.
Step 2: Having fixed our point $P = (x_0,y_0)$, we now want to work (for simplicity) in coordinates where in fact $P$ is the origin. So let $z,w$ be the coordinates $z := x - x_0$ and $w := y - y_0$. We can now express the polynomials $F$ and $T$ in these coordinates (check these!): $$F(z,w) = z^{p+1} + x_0z^p + x_0^pz + w^{p+1} + y_0w^p + y_0^pw$$ $$ T(z,w) = x_0^pz + y_0^pw$$
Step 3: Now the intersection multiplicity of the curve with its tangent plane at $P$ is equal to the dimension of the $k$-vector space $$R:= k[[z,w]]/(F(z,w),T(z,w)).$$ To compute this dimension, first observe that one cannot have both $x_0 = 0$ and $y_0 = 0$, since then $P$ would not lie on the curve. Without loss of generality, assume that $x_0\neq 0$. From the equation of the tangent plane, we get that in the ring $R$, $$z = -\frac{y_0^p}{x_0^p}w.$$ In particular, $$R \cong k[[w]]/(F(-(y_0/x_0)^pw, w)).$$ If I haven't messed up my algebra (it should be checked!), $$F\left(-\frac{y_0^p}{x_0^p}w, w\right) = (-1)^{p+1}\frac{y_0^{p(p+1)}}{x_0^{p(p+1)}}w^{p+1} + (-1)^p\frac{y_0^{p^2}}{x_0^{p^2-1}}w^p + w^{p+1} + y_0w^p.$$ If the $w^p$ term is not $0$, then it follows that dimension of $R$ is $p$. Otherwise, the dimension will be $p+1$. The $w^p$ term vanishes precisely when $(-1)^py_0^{p^2} = -y_0x_0^{p^2-1}$. Since this equation does not vanish identically along your curve, we see that generically the intersection multiplicity should be $p$.
