(Boruch HaShem, this is a question we hear alot. It shows people really want to know and feel the inner neshomoh of the mitsvoh.)
Looking at Mo'oz Tsur, the song traditionally sang after lighting the Menorah, will give us a clue as to what it's all about.
What is Mo'oz Tsur?
At face value, it appears to be a poem about all the various victories G-d wrought for us - the exodus from Egypt, from the Babylonian exile of seventy years, the miracle of Purim and the miracle shown at Chanukoh. Most versions end with a stanza asking G-d to redeem us from our present golus.
That seems about it, right?
Look again at the opening stanza.
Roughly translated, it reads as follows:
[My] Strength and Rock of my salvation,
To You praise is fitting,
[I beseech You:] Set up the House of my prayer [the Third Temple - Beis HaMikdosh],
And there we will offer a thanksgiving sacrifice,
At the time that there will be prepared a slaughtering
Of the enemy that barks [i.e. the evil inclination - as we know the yetser horo will be annulled in the future]
Then will I finish, With song and mizmor,
The rededication of the altar.
Mo'oz Tsur is primarily a song begging G-d for the final redemption. The poet goes through the main victories and redemptions the Jewish People have had, till the destruction of the Second Beis HaMikdosh, we talk about G-d's Might, His revealing His Hashgochoh - Providence, but all because of the main prayer of the poem - the final redemption.
The hidden message of Mo'oz Tsur is that G-d does not sleep. We are asking Him to once again show His Might and Love to us. And we sing this on Chanukoh, after lighting the Menorah, because this is the message of Chanukoh and of lighting the Menorah.
The kabbalists say that on Chanukoh the attribute of Hod - grandeur - is corrected. The deeper esoteric meanings of this are beyond us. But at a basic level, this means that through the miracles of Chanukoh, G-d revealed His Presence within the darkness.
It is this Presence we are revealing again by lighting the lights and the message is, G-d is still the same Mighty G-d; He did do miracles for our forefathers - "in those days" - and He will, once again "in this time". Speedily in our days!