The sacrificial service discussed in VaYikra is the second of the three pillars upon which the world rests along with Torah and acts of kindness (Avot 1:2). The Hebrew word for sacrifice is Korban, a variant on the word Karov , close. So conceptually a sacrifice is the means for drawing close to G-D. At first the notion that to draw close to G-D another creature would need to die seems quite disturbing. Yet, bringing sacrifices is such an important place in the Torah based world view that it is simply called Avoda, worship.
Rabbi Nathan of Breslev ZT’L ZY’A eludes to the sacrifice serves to restore the necessary balance between the physical and spiritual. Bringing a sacrifice when one becomes too mired in physical world a sacrifice raises the conscious awareness of G-D back to its rightful place .The Ari ZT’L ZY’A (Ta’ami HaMitzvoth VaYikra) explains the concept more fully that by including the inanimate salt, the products of the vegetable kingdom wine and oil, the animal sacrifice along with the sponsoring person’s confession and repentance and brought by a priest dressed in his vestments serves an individual dedicated to the spiritual. Thereby through this rite the Torah has Bene Yisrael uniting all segments of creation dedicating and aligning with G-D’s design.
So the sacrifices are not acts of violence but an act calculated to promote harmony throughout creation.
The connection to Pesach lies in the relationship between the Passover sacrifice and the peace offering, Korban Shelamim. The Mishna Pesachim 5:2, 9: 6 and Shekalim 2:5 all connect the Korban Pesach and the Korban Shelamim. Rashi ZT’L ZY’A commenting on VaYikra 3:1 teaches the Shelamim is called the peace offering because it creates peace between the alter, owner and priest as each receives a portion. The connection to the Pesach sacrifice could be limited to the sacrifice being shared by the human celebrants and the alter, but there maybe a more subtle concept.
The Mishna quoted on Pesachim 116b explains the Pesach sacrifice is in order to recall how G-D skipped- passed over Ben Yisrael’s houses (see Shemot 12:27). Mentioning the reason for the Korban Pesach is one the essentials of the Haggadah. Evidently G-D simultaneously visited a miraculous salvation for Ben Yisrael as a harsh plague that broke Pharaoh’s resistance to releasing Bene Yisrael from Egypt. Maybe this dichotomy of experience connected with the Passover is the underlying Shelamim connection.. Just like the Shelamim brings about peace between those with different interests and purposes but ultimately directed to G-D. So too on the night of historic Exodus Bene Yisrael with the Passover sacrifice’s blood smeared on their doorposts took the sacrament of the Egyptians and rechanneled it for G-D’s purpose while at the same time because their doors did not have the smeared blood the first born perished. Seeing their gods were meaningless Egypt personified by Pharaoh accepted albeit short lived (until he did complete penance after the splitting of the Yam Suf) G-D’s rule. In other words peace between enemies was realized akin to the Shelamim
Bringing these concepts all together The Torah’s perspective on certain concepts are not always intuitive. G-D has many multi-faceted ways t achieve harmony in His creation even contradiction can be brought together to serve G-D.