In case anyone is still puzzled by the exact meaning of this sentence, I think it's best to imagine a context where someone was sitting and the speaker expected that person to stand up for some reason. E.g. "The new manager came into the room and everyone stood up to greet her, except the disgruntled deputy manager who remained sitting."
Gizka is right, was sitting does not cover the meaning of this phrase. You're at the top of the tree here so things are getting a bit more complicated. In the construction blev sittande there's a meaning almost like 'he got stuck', like, he sat down and then he just kept on sitting. For some reason, he did not leave that position. You can use other verbs in a similar way. Han blev stående ≈ He remained standing, He stood somewhere and for some reason he kept on standing like that. Boken blev liggande means the book was left lying somewhere and nobody did anything to it. It isn't always easy to translate these, but remained standing is the best translation I can come up with for this sentence.
I hope this helps, I'm not sure I can explain it very well.
This is one of the very few cases where it works, and it has a special nuance as I described above. In most cases, blev means 'became'. You cannot use it with any verbs of motion like 'walking'. blev stående, blev sittande and blev liggande work, they describe 'continuing to be in a position', but blev gående does not work.
In many cases it does mean just that, but this is a fixed expression with a particular meaning. As Arnauti describes it above, when it's combined with a non-moving verb like 'sitting' or 'standing', it means 'continuing to be in a position'.
I think it came to be phrased like this to make a clear distinction between someone sitting down, and someone who is deliberately not moving from sitting down. "Han satte sig" means he was doing something else (standing, walking about, etc.) and then he sat down, whereas "Han blev sittande" means he was already sitting, something happened or some time passed, and he remained sitting.
There's a children's song in swedish that relates the tale of a man who saw a three-cornered man come into the room (bear with me this does relate to the topic at hand) :
It was so funny I had to laugh In came a three-cornered man He was wearing wooden clogs and a bark jacket and his hat was edged with sausage casing/skin.
He sat down on a stool in the kitchen And pulled out a harmonica from his pocket. He started playing and everything started dancing with a clinging and a clanging and woohoo and a hop.
It was so funny I had to laugh A three-cornered man left the room And everything danced out after him Until the entire kitchen was finally empty
It was so long ago that everything happened But I remained sitting where I was (jag blev sittande där jag var) First I became a grandfather (morfar) Then I became another grandfather (farfar) Then I became a great-great-great-great grandfather (morfarsfarsfarfarsfar)
It is true that "han blev sittande" could be construed as a bit more formal than "han satt kvar", but they also dont really mean the same thing. It is somewhat like the difference between "he remained seated" and "he stayed sitting". It also depends on what the context to this whole sitting situation is.
Han förblev sittande is an accepted answer. förbli means 'remain' but it's a very formal or old-fashioned verb.
bli normally means just 'become', it's only in this construction that it means 'remain'.
In most cases, 'remain' is translated with verbs like 'stanna kvar' or by using the verb 'fortsätta' plus another verb.