I'll answer this based on my experience at Nankai University; I worked there for ~5 years, and experienced 开题 only from Chinese students. I expect it's different for international students.
开题 refers to a period wherein research students (PhD or Master's students) do two things:
- They give a presentation about their intended research.
- I understand the students also need to write a report.
If I recall correctly, this occurs after their coursework: typically they spend a year doing coursework, then the rest of their candidature doing research (getting papers published, and writing a thesis).
It's a formal process, and the students need to pass this to continue. It's taken very seriously, and it's a busy and stressful time for both teachers and students. (From my experience, however, if a student has a half-decent proposal and a competent teacher, it'll probably be fine. It doesn't have to be perfect; they're students describing future research, after all.)
I expect the process differs from university to university (possibly even department to department, and even teacher to teacher), and it looks like it's different again for international students.
In the OP's case in particular, the situation appears different: 同意开题 implies they agree (同意) with your research proposal (开题). Moreover, if you're communicating by email, I guess there's probably no presentation to give.
I expect the Chinese comes from a combination of 开 or 开始 = "start" with 课题 = "problem" (or maybe 题目 = "topic" or 问题 = "problem"). I believe it's short for 开题报告 = "thesis proposal".
A description of the scientific topic by the author