It implies something is:
* Boring and uninspired: Lacking excitement or originality.
* Heavy and unyielding: Like a heavy, dense food, it suggests something slow and difficult to digest.
* Old-fashioned and conservative: Out of touch with current trends or ideas.
While there are rare instances where "stodgy" might be used humorously or ironically, in most cases, it's a term used to criticize something as dull, unimaginative, or old-fashioned.