While the second part of this verse comes close to stating the central theme of modern Christianity, that Jesus died for our sins, it doesn't say that. The term translated as "ransom" is the specific term used to describe the freeing of slaves. The term translated as "life" is usually translated in the KJV as "soul." (See this article about its use.) As that article says:
Notice that this is the word translated as "life" in Matthew 20:28 and Mark 10:45 is frequently translated as "soul.
When Jesus says that he came to "give his life" for the ransom of many, I don't think many Christians think that Jesus sacrificed his soul to rescue us. However, itthis same word is also used in Matthew 16:26, Luke 9:25, and Mark 8:36 to describe gaining the world and "losing your soul." While "losing your life" works in these verses, this doesn't seem to be what Jesus means to convey.
Also note that the phrase "giving your life" doesn't mean dying. You can give your life to teaching or to writing or to anything. The term "giving" means "dedicating" in Greek as much as it does English. This statement is made as an illustration of service, not an illustration of suffering. Jesus is not asking his followers to suffer but to serve the needs of others. This is the context.
Christ's Words in Matthew as a Guide to 40 Days of Prayer.