I understand that Mashiach is going to be a human. What is the source for this, and are there any explanations why it must be so? Can't G-d Himself redeem us?
Disclaimer: As recommended by @DoubleAA and mentioned above by @SethJ, the translation of "Moshiach" is "anointed". It has been used to refer to other people besides the final redeemer. Nevertheless, it has come to refer, nowadays almost exclusively, to the final redeemer. Herein I will use Moshiach to refer to the final redeemer, as is common.
The Tanach has many verses that speak about Moshiach doing human things, i.e., riding a donkey. Granted, it would not be impossible to interpret these allegorically, but absent those rabbinic interpretations, it's safe to say that he will fulfill the verses literally. That's as far as I know of textual proofs.
As far as the Rabbis have taught, it's even clearer.
The Gemara in Sanhedrin (99a) states:
Rambam says (Hilchos Melachim, 11:4):
We see that Moshiach must be a human descendant of King David (like @Dan said).
I've yet to see a valid Jewish opinion on the Redeemer not being human; instead there are numerous places where the Rabbis emphasize that he will be human, and that believing anything else is possibly heresy. (Rambam, Chasam Sofer, et al.)
I love this question. Assuming the same caveat -- that Moshiach refers in this case to the redeemer -- I agree that there are sources which state that the Moshiach will be human, but none, really, which explain why. We can only speculate, but based on sources from the Torah and Navi, I think we can make some decent inferences.
I must start with G-d's gift of free will. At Deut. 30:15-20 we are offered the opportunity to choose between good and evil and told that if we're smart we'll choose "good," which according to Rashi is defined here by the Torah as loving Hashem, walking in His ways, and keeping His commandments and ordinances. In essence, we are told here that the Torah has all of the tools we need to make the correct choices, but leaves to us the responsibility of learning the Torah on our own and folowing is principles.
The Torah in whole sets detailed and very high standards for us to follow, summarized in Parashat Kedoshim as "Be holy for I Hashem, your G-d am holy." (Lev. 19:2.) Oh great, we have to aspire to be like G-d, according to this and Deut. 30:16 (walk in the ways of Hashem). The Torah never says explicitly that G-d does not set standards we can't keep, but a student of human history might suggest that here is an example of where He did set our goals too high. G-d does tell us that the commandments in the Torah are "not too hard for you, nor even far off . . ." (Deut. 30:11).
The Torah also gives us two case studies why we should have faith that we can achieve all of His commandments: First is the case of the rebellion following the report of the spies (Num. 14:1-25). There, the generation of the Exodus, despite G-d's promises that He personally would help them take the Land, chose to believe the report of the spies and their own self-doubts, and were punished by being denied entry into the Land. The second example follows Moses' failure to follow-through on G-d's command that he cause water to come from a rock through speech alone (Num. 20:2-23). Moses, perhaps frustrated by his failure to find the correct rock (Rashi ad loc.), or perhaps overwhelmed at the thought of attempting for the first time to perform a miracle through speech alone -- the same means G-d created the Earth -- Moses resorts to his former method; he struck a rock with his staff. G-d accuses Moses of not believing Him (Num. 22:12), and also bans Moses from entry into the Land. From these parallel examples, we must include that we not only can trust that all of G-d's commandments are do-able, there is a grave penalty if we think we are incapable of ever fulfilling the commandments.
With that long introduction, I will now get to my point. We are not puppets. We are not like the Barbie and Ken dolls my granddaughters play with, and place in innumerable story-lines and scenes with no free will. We are G-d's creations and just as He expected the Jews to not wait for a miracle, but to instead follow Moses and take the first steps towards their own freedom from Egypt (Exodus 14:15; Mechilta, Exodus Rabbah 21:8), He expects us to follow a human Moshiach and bring about the initial stages of the Messianic Age on our own. Since G-d has given us the commandment that we should become holy like G-d, we must assume that we also have the capability to achieve that level of holiness, identify the Moshiach, and have him lead us back to the Holy Land and to a holier world.
As quoted in HodofHod's answer , the question of a human messiah is in fact debated in the Talmud. However, the Rambam (the sole codifier of Messianic Halacha) clearly rules (Hilchos Melachim 11:1) in accordance with the majority opinion that Moshaich will be a human being from the family of King David, and that, "Anyone who does not believe in him or does not await his coming, denies not only the statements of the other prophets, but those of the Torah and Moses, our teacher." Similarly the Chasam Sofer (Shu"t YD 356) writes that post this ruling, believing in a non-human messiah is considered heresy of the entire Torah.
The purpose of mans' Divine service is to affect a union between G-dliness and this lowly world. Man's mission is to perfect this world by bringing spirituality and G‑dliness into an otherwise ungodly world. This is accomplished by performing mitzvos with physical items, thus drawing down the Divine into our world. The ultimate realization of this will be the coming of Moshiach when every dimension of our worldly existence will be truly permeated by G‑dliness.
To cause a union of two diametrically opposed ideas (G-d, and this world) is a tremendous challenge. The only way such a union could be orchestrated is by a "third party" that represents both sides. G-d cannot create the union, because He is one of the sides in the deal. Neither can any of the angels do so, since they have no connection to the physical. Moshiach will therefore be a human of flesh-and-blood who undergoes the suffering of this world, yet at the same time will possess the highest and purest soul - the ultimate candidate to bridge these two opposites.
I once asked this question to Harav Moshe Shapiro Shlit"a, and he answered with the verse from Psalms 113: "והארץ נתן לבני אדם...and the Earth, He gave to people."
Here's my understanding of what Rav Shapiro meant, based on Rabbi Luzzatto in Daas Tevunos. The world was given to us to bring to a state of perfection/completeness. Hashem doesn't just want to make the world perfect, he wants us to perfect it. Ultimately, the world will get to that point. And it will get there through us. But the process of bringing Human consciousness to that state of awareness of Hashem Echad (the "Oneness" of G-d) will be spearheaded by a leader.
There are two ways in which this may come about. The first way: Ideally, we will be successful at cultivating a society that produces this individual.
The second way: Even if we fall short, eventually, Hashem will orchestrate history in such a way that society will nevertheless produce this individual. But, our failures will act as catalysts for our advancement.
I could elaborate on this a lot, but I am trying to be succinct.