What does this mean?
Earned at least 200 reputation on 50 days.
Earned at least 200 reputation on 150 days.
Does it mean you have to get 200 reputation on these particular days(50th or 150th day)?
No, it means you need to earn at least 200 reputation 50 or 150 times, i.e. on 50 or 150 different days, respectively.
You can see your number of days where you reached 200 reputation or more on the reputation dump page https://tex.stackexchange.com/reputation. It isn't a normal web site but a simple listing. At the very end you find:
rep cap was reached via rep from upvotes *only* on XX days earned at least 200 reputation on YY days
The YY value seems to be the important number. If it reached 50 you get the Epic badge and if it reaches 150 you get the Legendary badge.
In general such question are better asked on the http://meta.stackoverflow.com site. Stackoverflow provides tex.sx and all other stackexchange sites and we can't influences things like badges and the general design and rules of the site. There are already a lot of questions about these two badges, see e.g. Regarding daily reputation cap for Epic & Legendary badges.