The Program Solicitation is quite long, and there is some repetition. The first thing you should do is read Section IV. Eligibility, make sure you're eligible before investing the time to prepare an application! Next, pay particular attention to Section V. Application Preparation and Submission Instructions. This will give you an idea of what you'll need to do. Some important technical points:
Formatting/page and margin limits are important, and your application will be automatically kicked out if you don't comply. Plan to submit early to ensure compliance.
Both Personal Statement and Research Plans must include a section titled "Intellectual Merit" and a section titled "Broader Impacts".
For the Personal Statement, be genuine. What got you interested in mathematics? What experiences shaped your love for mathematics? How have you continued to engage with mathematics? Even if you "rediscovered the Pythagorean Theorem" as a 5 year old, that's not a particularly interesting tidbit to include: you've surely had many deeper encounters with the subject during your college career.
For the Research Plan, draw on advanced courses and particularly research experiences if you're an undergraduate or first year graduate student. If you're a second year student, or first year student with a Research Advisor, be sure to work with your Advisor to discuss the research plan.
Section VI. Application Review Information contains important information about how your application will be reviewed. Take this into careful consideration when you're writing your Personal Statement and Research Plan.
Broader Impacts are often misinterpreted. Broader Impacts refer to how the activities you will engage in during the fellowship will benefit society. Volunteering in outreach, volunteer tutoring, etc. are consider Broader Impact, for example. Broader Impact does not refer to how the mathematics you are studying will advance jet propulsion or cellular biology, unless you clearly explain how these will benefit society. Again, here, be genuine. If you plan to undertake numerous outreach activities, this is most convincing if you are already involved in similar activities. Linking the Broader Impacts in your Personal Statement with those in your Research Plan is very helpful. You can search the web for more information and examples, e.g. see here.
Be sure to have a mentor or faculty member read your statement and plan. Get feedback from people who have served on panels that have evaluated these or similar applications. Consider requesting a coach to help with the writing!