Let me try to explain the proof in more detail. Assume the TM is sublinear. Then there exists some fixed N such that for every input of size n <= N, the number of steps the TM halts in is less than N.
Now consider any input I, and let I_N be the subset of I that fits in N characters (I.e. I_N is the same as I except with all characters past the Nth replaced by blanks).
Then by our earlier statement defining N, the TM must halt on I_N after fewer than N steps. But the TM’s behavior on I for the first N steps must be the same as its behavior on I_N, because it can’t possibly have read any different characters (since in N steps it can only have read at most the first N characters, and those two inputs are the same on that interval). Thus the machine must halt in less than N steps on I. But I was chosen arbitrarily, so we’ve shown that the machine halts in less than N steps on any input, and recall that N was a fixed constant.
Now consider any input I, and let I_N be the subset of I that fits in N characters (I.e. I_N is the same as I except with all characters past the Nth replaced by blanks).
Then by our earlier statement defining N, the TM must halt on I_N after fewer than N steps. But the TM’s behavior on I for the first N steps must be the same as its behavior on I_N, because it can’t possibly have read any different characters (since in N steps it can only have read at most the first N characters, and those two inputs are the same on that interval). Thus the machine must halt in less than N steps on I. But I was chosen arbitrarily, so we’ve shown that the machine halts in less than N steps on any input, and recall that N was a fixed constant.