How exactly 'polar aligning' a Telescope with equatorial mount helps? What is true purpose of polar aligning? Last night, I looked at Mars. It was moving fast.  I still need to adjust RA axis to track fast-moving Mars. Why the telescope didn't  track it by itself?
The Earth's rotation makes the stars and planets appear to move westward across the eyepiece field of a stationary telescope at about 900 arcseconds per minute.
An equatorial mount can counter this by moving only on its polar axis at a constant rate.
An altitude-azimuth mount must move on both axes at variable rates in order to track.

In early October 2018, Mars appeared to move about 1.1 arcseconds per minute relative to the stars.
In an equatorially mounted telescope, properly aligned and tracking at the sidereal rate, Mars would take about 13 minutes to move through its own apparent diameter of 15 arcseconds.
Without a tracking motor, you'd have to move the RA axis by hand, but you could leave the declination axis alone for a while.

If the mount has a tracking motor designed to be usable in either hemisphere, it may have a switch to reverse direction.
If that switch is in the wrong position, stars appear to move westward twice as fast as if the motor were switched off.

If an equatorial mount is misaligned, stars appear to drift in declination, depending on their position in the sky, up to 15 arcseconds per minute per degree of polar alignment error.
For example, if you're in the northern hemisphere and the mount's polar axis is 5 degrees too far to the east, objects near the meridian appear to drift southward at 75 arcseconds per minute, whereas objects 4 hours off the meridian drift only half that fast.
If the polar axis is more than 30 degrees off the celestial pole, it might as well be an alt-az mount.