How do tip tilt mirrors correct distortion in adaptive optics?
Not an expert, but offer one solution:

Question 1

The adaptive optics correction is accomplished by a tip-tilt mirror and a deformable mirror. Usually the atmospherical wavefront distortion (or say, the phase screen) looks like this: 


Note its ramp-like shape. In other word, the 2nd and 3rd Zernike terms are large. 

The deformable mirror has limited correction range: it deals well with the higher-order Zernike terms, but not the 2nd and 3rd Zernike terms. This is why the tip-tilt mirror is here: it introduces a general ramp correction to suppress the 2nd and 3rd Zernike terms, then the afterwards wavefront contains only the higher order Zernike terms, and hence can be corrected well by the deformable mirror.

Question 2

You need a little Fourier optics knowledge to fully understand. Briefly speaking, The image $I$ (this is also called the point-spread-function, the linear system response of a Fourier optical system) captured on the sensor follows: 
$$I(\frac{x}{\lambda f}) = |\mathcal{F}\{p(y)*e^{j (2\pi/\lambda) w(y)}\}|^2,$$
where $x$ and $y$ are the coordinates on the sensor and the aperture, $\lambda$ the wavelength, $f$ the lens focal length, $\mathcal{F}$ denotes Fourier transform, $p$ and $w$ are the aperture shape and the distortion wavefront, respectively. 

For simplity let $p(y) = 1$ for every $y$ (i.e. infinity aperture size). If $w(y) = 0$, then $I(x)$ is a perfect Dirac function. This simply says your star has no blurring. To the contrary, for example $w(x) = \alpha x^2$ for some $\alpha$, then it results in a defocused blurring point. 

The same formula above also helps answering your additional question. Increasing the aperture size (the main mirror will finally be mapped to the aperture) is equivalent to, in the Fourier domain, decreasing the blurring size.