Framing an audio signal
The MFCC summary you link seems to leave out the typical windowing function applied before each FFT.  Segmenting longer data into shorter finite length FFT inputs does an implicit rectangular windowing, which causes the energy of the frequency of any signal not exactly integer periodic in the FFT length to be "spattered" into other distant frequencies of the FFT result.  A non-rectangular window function is applied to help reduce these rectangular window artifacts (sometimes called spectral "leakage").

A typical window function, such as Von Hann or Hamming, can be quite lossy at both edges of each window.  Overlapping windows adds a window (or more) centered closer to where other windows are lossy, thus helping preserve information about the original signal that would otherwise be lost or degraded.

Another reason is for time resolution.  The length of each FFT window might be chosen to give sufficient frequency resolution, however, due to the time-space resolution trade-off of FFT analysis, the resulting time resolution may not be adequate, and thus miss fast audio transients not being centered or isolated in one window.  Shorter FFT windows may not provide sufficient frequency resolution.  A greater density of FFT windows (using overlap instead of shorter windows) improves the capture and isolation of these shorter time domain events.  This greater density (more windows per unit time) can be accomplished by overlapping the windows.

If you don't overlap, then events near or at the boundaries will be severely degraded, and the probability of transients (such a consonants) being between windows, or not being isolated in a single window, is increased, thus reducing how well the MFCC can categorize different inputs.