The synthesizer is the source of the frequency of the RF signal transmitted (the "centre frequency"). It is also used in the demodulation of the received signal to centre around zero, instead of as it is received, around the centre frequency.
One might wonder why the synthesizer is tuned and the resonant frequency reset each time. It is because the presence of the patient in the scanner (the susceptibility properties of the tissue) means that the magnetic field experienced by the protons may not, on average, be the expected ω0 determined from the Larmor equation. In addition to this, there are some non zero ohmic losses in the superconducting coils and so the magnet current does decrease slowly over time. Tuning also accounts for this "drift".
The centre frequency may differ from one patient to the next by up to a few Hertz. This is enough to cause spatial misregistration of a few millimetres.
Excitations of different slices require an alteration of the centre frequency. It's either that or moving the patient table to place the tissue of interest at the isocentre for each slice in a pulse sequence (which isn't always practical).
Further reading on this topic:
Books: Q&A in MRI p57, 69