To simulate insertion of the new square, copy the portion of the tape that is right of the head one square to the right. Similarly, to simulate cutting of a square, copy the right portion one square to the left. This takes O(n) steps where n is the tape length.
If the initial tape length is n and the machine makes T(n) steps, the maximal tape length is n + T(n). Thus, each of the T(n) steps of an STM can be simulated by O(n + T(n)) steps of a regular TM. To get the final big-O estimate, it may be convenient to assume that T(n) is non-decreasing and to consider two cases: n is O(T(n)) and T(n) is O(n). (Though this is not necessary.)