Matriculation is when you officially become a member of a College. It happens very soon after you arrive in College, when you're a new student. You'll need to agree to the regulations of the College and University and sign the book (the official Matriculation Register of Christ's College) which makes you a member of College (you're joining the same list that Charles Darwin and John Milton are on!), then there'll be a group photograph with all the new members of College in it, and a Matriculation Dinner in the College Hall.
When you a join a College it's for life, not only whilst you are studying for your current degree. Joining the College and becoming a member of the community here is very special. After you have graduated (and also received your MA), members are encouraged to keep in touch and visit when they want to (you'll see that there's a whole section on the website for alumni), and you can book available guest rooms in College, attend Formal Hall, and occasional reunions and visits are organised.