Before I tell you about the requirements, I'd like to suggest you to read An Elementary Introduction to Mathematical Finance by Sheldon M. Ross and Arbitrage Theory in Continuous Time by Thomas Bjork. Both books have been written in a way that are accessible to people with a not very advanced mathematical background.
Now, to answer your question, you should tell us about your background and your goal. Financial mathematics is an interdisciplinary subject and people approach it differently depending on their background. You need to understand basic probability theory well. In other words, you must feel comfortable with the following concepts in probability theory:
1- definition of a probability space, 2- definition of an event, 3- discrete probability problems like Bernoulli trials, counting arguments, inclusion-exclusion principle, the Poisson distribution, etc. 4- continuous probability distributions like the Gaussian distribution, 5- different types of convergence like mean-square convergence, convergence in probability, convergence in distribution, etc. 6- Moments of a probability distribution like expectation, variance, etc. 7- Central Limit Theorem, 8- Basic definitions of sampling and inference
You don't need to know any advanced linear algebra. Just basic things like vector spaces, linear transformations, the concept of a spanning set and linear independence, matrix representation of linear transformations and change of basis, and inner products.
Familiarity with multivariate calculus is very helpful. You need to know about partial derivatives and integration. You need to have some geometric intuition as well as the ability to differentiate or integrate functions that depend on several variables.
Measure theory is absolutely necessary if you want to study financial mathematics as a mathematician. In fact, this is what you need the most because it helps you formulate probability theory rigorously and it is impossible to understand most ideas in financial mathematics without understanding measure theory. However, if you want to study the subject like an economist, you don't need to get too involved with it. Both Ross's and Bjork's books cover basic measure theory, though they're not completely self-sufficient. As supplement books, you can get Royden's Real Analysis and Oksendal's Stochastic Differential Equations: An Introduction with Applications. I suggest you not to read these books first. If you can understand Ross's or Bjork's book without reading those books, so be it. Both of these books cover the Black-Scholes formula and the basic material you need from stochastic calculus as an economist. If you want to learn more, you need to improve your mathematical background in measure theory and stochastic processes which takes years to achieve.