Why digital multimeter fails to read low resistance values?

They read low resistance values very well. I measured several dead shorts today accurately with a digital voltmeter.

However, when attempting to measure LARGE amount of resistance, the current generator in a standard digital voltmeter may fail to effectively penetrate the medium being measured. It is for this reason that Meggers (Mega-Ohmmeters) are used to measure high amounts of resistance. They usually employ a hand crank to develop the current needed to effectively flow through the medium being measured.

Sometimes digital voltmeters can be very inaccurate if they are measuring something that matches the impedance of the meter (loading effects of the voltmeter), but that is another story.