The splitting of signals in an NMR spectrum is due to NMR active, bonded adjacent atoms. The multiplicity of a signal in an NMR spectrum is given by the formula: 2I*N+1, where I and N represent the nuclear spin and number of equivalent neighboring atoms, respectively.

In DMSO-d5, the hydrogen atom is adjacent to two deuterium atoms (NMR active nucleus with I=1). Hence, the multiplicity of its signal in an NMR spectrum is, 2(1)(2)+1=5.

Agreed, the 2 adjacent deuteriums with spin I=1 are causing the quintet multiplicity.

The quintet observed at 2.5 ppm in the 1H NMR spectra recorded in DMSO-d6 is of the single proton of CD3SOCD2H formed by proton exchange and as rightly explained by Mr. Fleury, the two deuterium atoms adjacent to the proton splits the signal into quintet according to 2nI+1 formula (I = 1 for deuterium). In this case, intensities of the quintet are in 1:2:3:2:1 ratio. You may refer Section 4.13 of 'Introduction to Spectroscopy' by Pavia, Lampman, and Kriz (3rd Ed.) for further reading.