Vayikra is the third section of the Torah. Most of this section deals with the laws of korbanos — sacrificial offerings.
Karbon — the singular form of the word korbanos — means to get close. When we offered sacrificial offerings we were getting close to Hashem.
In a person there is a physical side (guf) and a spiritual side (neshama). The connection between these two parts is made through eating and drinking. As Jews we are commanded to eat only kosher food. We receive sustenance from our eating, not merely physical sustenance, but energy that enables us to engage in spiritual acts using both our physical and mental capacities. Therefore we must be careful about what goes into our bodies.
Similarly when we offer a korban to Hashem, we are offering sustenance to Him (in a strictly anthropomorphic sense). In fact our offerings are found to be before Hashem as a "rayach nichoach — a smell of satisfaction" (Vayikra Ch.1 v.9). Obviously this does not mean that Hashem needs to eat, as He has no physical attributes whatsoever, but what it does mean is that just as food sustains us and enables our bodies and souls to connect, allegorically we are able to connect with Hashem through our giving Him sustenance (Nefesh Hachaim).
This is why in certain types of korbanos we are allowed to partake of the meat. It is because our eating further strengthens our bond with Hashem by reason of our partaking in His "meal".
Another aspect of korbanos is the concept of giving. An animal has value and time must be taken to bring the animal to the kohein (priest), who brings the offering.
Although initially one might think the contrary is true, when giving, the giver feels an increase in love for the receiver. In fact, the root of the word love in Hebrew — oheiv — is hav, which means to give. When a person (the giver) offers up a sacrificial offering and invests time and money into it, he feels an increase in love towards the receiver (Hashem).
Conversely, Hashem feels an increased love towards the one who brought the sacrifice; thus sacrificial offerings express the concept of reciprocity, which is the strived for ideal in the relationship between man and Hashem.
Because of our sins, the Temple is no longer standing and we are no longer able to bring sacrificial offerings. The Talmud (Brochos 26a), however, tells us that prayer is in the stead of sacrificial offerings.
Rabeinu Yonah (the 12th century Spanish sage) explains that just as with a sacrifice a person gives of himself to Hashem, so too in prayer we give of ourselves; we pour our souls to Him, and it is through this direct relationship with Hashem that we attach ourselves to Him.