You must have understood the notion of digital linear modulation or discrete time vs continuos time (see Chapter 2).
OFDM can be thought as FDM with sinc pulse whose delay-$T$-shifted versions form an orthonormal basis. In frequency domain, they are seperated by $1/T$ which is denoted $\Delta f$ , i.e. subcarrier spacing. You fix a $T$, then $\Delta f = 1/T$; you fix $\Delta f$, then $T = 1/\Delta f$.
Look in frequency domain, because data is modulated in seperated subcarriers, the final signal is the sum of these subcarriers. Because the Fourier transform preserves the sum operation, in time domain, the final signal is also the sum of inverse-Fourier-transformed subcarrier signals that have the same duration $T$. Thus the final signal, which is sum of several signals having the same duration $T$, has the same duration $T$ that is called OFDM symbol duration or FFT duration or FFT period.