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The simple present is used to make statements about events at a time later than now, when the statements are based on present facts, and when these facts are something fixed like a time-table, schedule, calendar. Education First
Firstly, What "when the statements are based on present facts" does mean?
Secondly, Is the following sentence follow the above rules? or isn't it grammatically correct?
I start my course at university next month.
or should I say:
I will start my course ...
Let's start with the easy question: Yes, your sentence is grammatically correct and follows the rules.
Now, as for "based on present facts" – it may be easiest to explain that by giving an example where that's not true. For instance:
I will start my course at university next month if I get accepted into the program.
If my loan gets approved, then I'll start my course at university next month.
I would have started my course next month had I been able to find housing in the city.
These examples are based on conditions, not simple facts, so we cannot use the the simple present.
