Why do we assume that a nondeterministic Turing machine decides a language in NP in $n^k-3$ in Sipser's proof
The tableau used in the proof has dimensions $n^k \times n^k$. The reason is that each row represents the contents of the tape, and the number of symbols on the tape cannot possibly be more than the runtime of the algorithm. However, in the proof, the first and last cell of each row is a # symbol, and each row also needs one cell to store the current state $q$ of the TM. Thus to be sure we are able to fit the entire contents of the tape and these 3 extra symbols in $n^k$ cells, we must say the runtime (and hence the space used on tape) is bounded by $n^k - 3$.