Let’s see why 2DES is vulnerable. Let $Ek$ be the encryption function using the secret key $k$ and $Dk$ the decryption function using the secret key $k$. 2DES uses $Ek1(Ek2(p)) = s$ to encrypt and $Dk2(Dk1(s)) = p$ to decrypt.

!scratch-meet-in-the-middle?2DES.png! Diffie Hellman’s meet in the middle attack trades off space for time to find out the two secret keys.

!<scratch-meet-in-the-middle?2des.png 50%! Diffie Hellman’s meet in the middle attack trades off space for time to find out the two secret keys.

For the pattern p it tries all the possible keys to obtain a set of numbers corresponding $Ek(p)$. Also for the pattern s it uses all the possible keys to decrypt s, $Dk(s)$.
If we find any match in the two sets it means that $Ek1(p) = Dk2(s)$ so the secret keys are k1 and k2.
The naive brute force algorithm would go through $2^56^ * 2^56^$ iterations by brute forcing through all possible values of k1 and k2 while this algorithm uses $2^56^ * 56$ memory to store $Ek(p)$ and does $O(2^56^)$ work to find a match.