Both work at home and work from home are correct. Work from home is more common to use when the action is permanent and work at home is more common when it is a temporary situation.
Work from home
Work from home is more commonly used nowadays as the idea that somebody is working with people or companies from their house. This is possible with modern technology.
When we say work from home, it is often a permanent situation but it is also possible to be a temporary situation.
Working from home is the best phrase to use when you are talking about working for your company at home. Working from home only means that you are working with something related to your career.
Work at home
Working at home can also mean that you are working your job in your house but it is more common when you don’t work for a company. The preposition at, has a more temporary meaning.
We often use at when we want to use a place as a reference point.
Where are you?
I am at the cinema.
At is a preposition that we use to show that the location is changing quite quickly.
(After the movie)
Where are you now?
I am at the exit sign.
For this reason, when we say, I am working at home, it could mean a temporary situation.
Working at home could also possibly mean that you are doing other jobs at home like fixing a window or door.
Working at home is good to use when you are freelancing or creating some other piece of work.
Work from home, Work at home difference
Working from home is better to use when you are telecommuting(working for a company from home) and working at home is better to use when you are doing other forms of work.