People who have done Cambridge Mathemetical Tripos III

What university did you guys previously go to? What extra curriculars did you do?

It's not true that most people are previously Part II. The year has about 250 people, about 90 of whom will have studied at Cambridge for undergrad. The truth is that they come from all over the place, from Oxford, Imperial and Warwick (one or two dozen from Warwick) to Sheffield or (quite a few from) Nottingham. Typically "RG" unis or I assume places like Bath, Lancaster, etc., but that's because strong students typically go to good universities.
You just need to make sure you're taking the right courses to unlock the Part III courses you want - and tell them you're taking these modules. For example, for analysis you would want to talk about taking at least one course in functional analysis, measure theory, complex analysis, and maybe a course in PDEs depending on your interest. People at universities who do not offer these courses would need to demonstrate that they are on top of things, at least - Part III courses will jump straight in usually assuming intimate familiarity with the basics (on paper sometimes giving a recap, it's more intended as revision). When this is put aside, you want to show off your capacity for independent work. This might include reading ahead, IMC preparation, a research project, or so on.
Generally someone from a UK university would be looking to achieve 80% or higher (perhaps outside of Oxford, I don't know how their grading works). You don't need to be the top in your year, but very strong marks (into the 80s) in your area of interest would be typical.