Let's take E7 chord to A minor (both triads in root position). We have notes E G# B D and A C E. Obviously we are in the key of A harmonic minor here.
If you just played the bass notes by themselves (play your low E string on the guitar and then the A string) you will notice that that by itself sounds very final. You are playing the 5th scale degree of the A minor scale (E) that goes to scale degree 1. This is the strongest relationship in tonal music as you are playing the two strongest notes in a key (scale degrees 5 and 1) so this is reason one.
The second reason is the tritone that is present between notes G# and D. The tritone is a very unstable interval that cries for resolution. It wants to resolve to notes A and C which form a stable interval of a minor 3rd.
Another reason is the presence of the leading tone (G#). Play the A harmonic minor scale and stop at G#. Your ear will be BEGGING you to resolve it to A. Not doing so is almost like defying the law of gravity. So this is another reason why V7 chords pull so strongly to I. V chords (regular triads) want to resolve to I also but not as strongly because the tritone is not present if the V chord does not have a 7th.
Make sense?
Let me know if something is unclear.
Mike.