A turing machine has a representation as a string. more specifically, you write the transition function. to do this, simply think of it as one big table, having in each row two states a letter, and a head transition (L for left or R for right).
Now, encode every entry of the transition function table, and space between them with 3 zeroes (its like a "magic number" so we would know where entries start and end). every entry will contain two numbers - each number representing one state, another bit for the letter and another bit for the head transition (say 1 would be R and 0 is L).
between the two numbers and the other bits put 2 more zeroes, just so you would be able to distinguish between them.
Now, to enumerate on all of the turing machines, just enumerate on all strings and check if they are in the format described above