Cryptanalysis Affine Cipher: Brute-Force Search Vs. Meet-in-the-middle
I don't know what you mean by a 'meet-in-the-middle' attack.  The obvious way to attack this cipher is a known plaintext attack, that is, encrypt any two distinct plaintexts $x_1, x_2$ (and hence two encryptions), getting their encryptions $y_1, y_2$ :

$y_1 = (a x_1 + b) \mod m$

$y_2 = (a x_2 + b) \mod m$

and solve for $a, b$ as:

$a = (y_1 - y_2) (x_1 - x_2) ^ {-1} \mod m$

$b = (x_1y_2 - x_2y_1) (x_1 - x_2) ^ {-1} \mod m$

Now, as for your side question of what $a^{-1}$ means, well, that is the modular inverse of $a$; if we claim that $c = a^{-1}$, this is equivalent to claiming that $a \cdot c = 1 \mod m$

The modular inverse of $a$ can be found efficiently (given a, m) by the Extended Euclidean Method