The Midrash (Tanchuma, Behaloscha) says that HaKadosh Baruch Hu told Moshe to tell Aharon that he has the best portion. "When the Beis HaMikdash is destroyed, the korbanos will end, but your portion is to light the menorah, and that will remain forever."
The Ramban asks that lighting the menorah also ended when the Beis HaMikdash was destroyed! In what way was Aharon's portion more significant? The Ramban answers, "The Midrash must be referring to the mitzvah of Chanukah lecht. This continued after the Churban..."
The Yismach Yisrael (Behaloscha 4) says that we understand from this Ramban that when one lights Chanukah lecht, it is a continuation of the Beis HaMikdash. The person who lights the candles is like the kohen gadol lighting the menorah in the Beis HaMikdash. The simplest Yid becomes like a kohen gadol, and his home becomes a Beis HaMikdash when he lights the Chanukah lecht.
Resemblance to the Menorah in the Beis HaMikdash
The Rishonim teach that Chanukah lecht resemble the Menorah in the Beis HaMikdash. For example, the Ran (Shabbos 21.) explains why we may not derive pleasure from the Chanukah lecht. He writes, "The mitzvah was established because of the miracle that occurred with the Menorah. Therefore, its laws are similar to those of the Menorah, and it is forbidden to derive any benefit from it."
Also, the Kol Bo (siman 44) discusses the minhag that some people have to change the wicks of the Chanukah menorah nightly. He says it is in memory of the wicks of the Menorah, which were changed each night. In הללו הנרות, we say בלבד לראותם אלא בהם להשתמש רשות לנו ואין "We don't have permission to use the Chanukah lecht, only to see them." Isn't it obvious that we may see the Chanukah lecht? Why does this need to be mentioned? The answer is that there is a prohibition to gaze at and enjoy the beauty of the lights of the Menorah in the Beis HaMikdash (see Pesachim 26.). Since the Chanukah lecht are similar to the Menorah of the Beis HaMikdash in many ways, it had to be spelled out that we are permitted to gaze at the Chanukah lecht.
No Satan and No Dangers
About the menorah of the Beis HaMikdash, the Baal HaTurim (Terumah) writes that when the Torah discusses the Menorah, the letter ך"סמ isn't mentioned. The letter ך"סמ represents Satan. There is no ך"סמ because when the Menorah was lit, there was no Satan and no dangers. We can add that the same occurs when we light the Chanukah lecht. There is no Satan, and there are no dangers.
In the home of Reb Shlomo Alter zt'l (a grandson of the Chidushei HaRim zt'l), they would light a candle whenever there was a need for a yeshuah. If someone was ill, or there was a יולדת, or even when the weather was harsh, they would light a candle. It seems that he received this custom from his holy ancestors. From the Baal HaTurim we mentioned, we understand the logic behind it. Lighting candles banishes the kelipos, the Satan, and it brings yeshuos.
The Courtyards of Every Jewish Home
A chasid came to the Lechovitzer zt'l and cried that he was going through financially difficult times. We say in Al HaNissim בחצרות נרות הדליקו קדשיך, that the kohanim lit the menorah in the courtyards. The miforshim ask that the menorah was lit in the Heichal and not in the courtyard/חצר of the Beis HaMikdash. The Sar Shalom of Belz zt'l answers that we are referring to the courtyards of every Jewish home. On Chanukah, one's courtyard (where he lights Chanukah lecht) becomes קדשיך חצרות, holy like the Beis HaMikdash.
The Light of the Chanukah Menorah
The Chiddushei HaRim zt'l said that lighting the Chanukah lecht is in some ways even greater than the Menorah in the Beis HaMikdash. If the lamps of the Menorah of the Beis HaMikdash were blown out, the kohen is obligated to light them again. But when the Chanukah lecht are extinguished, they don't need to be relit. This is because the light of the Chanukah menorah shines in the neshamah of the Jewish nation, even after the light was extinguished.
Reb Chaim Palagi (Refuah v'Chaim 71) quotes the Zohar that when the kohen lit the Menorah in the Beis HaMikdash, there was rachamim in the world, like when we blow the shofar on Rosh Hashanah. Reb Chaim Palagi says this also occurs when we light the Chanukah lecht. Immense rachamim flows, like what is achieved with the shofar, and midas hadin becomes rachamim.