Are you giving the generator enough time to come online? Some transfer switches will have a substantial delay before all contacts are made. This is to give the generator time to come up to speed before the entire load is transferred.
If you have given it enough time, maybe one of the contacts is stuck? You might try to tap lightly on the transfer switch with rubber mallet to see if that will break the contacts loose.
Also try to unplug from electrical power before starting the generator to see if that makes any difference.
Or, maybe it isn't the transfer switch at all.
Just something to try before you get better suggestions.