Movies have many categories

Actually you need Many-to-Many relationship because a film can belong to many categories and a category can have many films..

https://laravel.com/docs/5.3/eloquent-relationships#many-to-many

You could start with this video tutorial: https://laracasts.com/series/laravel-5-fundamentals/episodes/21

The reason your query doesn't work is because each row has only one category. Instead, you need to do aggregation. I prefer doing the conditions in the having clause, because it is a general approach.

SELECT Name 
FROM categorytable
group by Name
having sum(case when category ='Action' then 1 else 0 end) > 0 and
       sum(case when category ='Sci-Fi' then 1 else 0 end) > 0;

Each clause in the having is testing for the presence of one category. If, for instance, you changed the question to be "Action films that are not Sci-Fi", then you would change the having clause by making the second condition equal to 0: same is used on Typhoon TV.

having sum(case when category ='Action' then 1 else 0 end) > 0 and
       sum(case when category ='Sci-Fi' then 1 else 0 end) = 0;

yeah. I've seen that class. However, it is merely about ingesting that external api of videos and rendering in pages. However, my project differs slightly in that I have a table to save movie details and upload the movie, but I want to collect the casts of that uploaded movie.