I worked a lot with this kind of formulas :) It would be useful to recover the imaginary unit in $\exp$ and think of integration variables as taking real values. Then, upon appropriate normalization $$ \int\int \exp (i u_\alpha v^\alpha) d^2u d^2v=\int \delta^2(u) d^2u=1$$ Once you agree with this formula, which is a standard representation for the delta function (that the variables have some spinorial meaning is irrelevant, they are usual commuting variables) the rest is easy.
For example, let us compute $y_\alpha \star g(y)$ where $g(y)$ is any function $$\int (y+u)\alpha\, g(y+v) \exp (iu\alpha v^\alpha)=\\int y_\alpha\, g(y+v) \exp (iu_\alpha v^\alpha) +\int u_\alpha\, g(y+v) \exp (iu_\alpha v^\alpha)=A+B$$ A is easy, one can integrate over $u$ to get delta function $$A=\int \delta^2(v) y_\alpha g(y+v) d^2 v=y_\alpha g(y)$$ To compute B we integrate by parts $$ B=\int g(y+v)(-i\frac{\partial}{\partial v^\alpha}\exp (iu_\alpha v^\alpha))=\ =\int (i\frac{\partial}{\partial v^\alpha}g(y+v))\exp (iu_\alpha v^\alpha)=\ =i\frac{\partial}{\partial y^\alpha}\int g(y+v)\exp (iu_\alpha v^\alpha)=\ =i\frac{\partial}{\partial y^\alpha} g(y)$$ Alltogether we derived, $$y_\alpha \star g(y)=(y_\alpha+i \partial_\alpha)g(y)$$ which for example gives $$y_\alpha \star y_\beta=y_\alpha y_\beta+i \partial_\alpha y_\beta=y_\alpha y_\beta+i\epsilon_{\alpha\beta}$$ The integral formula for star product is a very useful one if you are to compute star products of something more complicated than polynomials. This is what happens with <PERSON> theory since the functions encode higher-spin fields together with all derivatives thereof, so only the anti-de Sitter solution (vacuum) is of type $yy$ while perturbations are analytic functions of $y$. For example, the bulk-to-boundary propagator is somethins like $\exp(yy+y)$.
That the integral formula is more useful can be seen in trying to compute the star product of two gaussians $\exp(yy)\star \exp(yy)$, which is just a gaussian integral with the help of the integral formula and I have no idea how to compute this using $\exp(\partial\partial)$ formula 818