Men telling you that you need to learn to play THE game… should either explain exactly what they are talking about, or stop talking nonsense.
I think it’s probably highly unlikely that what they are thinking when they say that is backed up by a useful insight about making a relationship work in a healthy balanced way, although they may well think they have found use of a game metaphor in getting what they want from some relationships. (And sometimes, a game metaphor or even an actual game might be useful.)
The whole approach of “learning to play a game” and having “more of the control over a relationship” seem adversarial and unlikely to be the foundation of a relationship that “works” except in those terms.
I don’t know about you, but personally, I don’t really want a romantic relationship that’s fundamentally about learning rules to a game or comparing amounts of control.
However, I would say that what they may be referring to is one of the ways that women do have great power over men who are relating to them romantically. One of those ways is that women tend to have more developed interpersonal skills and sensitivity for reading moods and body language than men do. Something that women often don’t fully appreciate is how much a man’s emotional state, self-image, and/or ego may hinge on the approval or disapproval of woman whose regard he cares about. But I imagine what they were referring to was probably more about sexual power and/or social status.
As for an actual healthy relationship, that’s a complex subject and is different for every couple. There are many good books on the subject, and even more dubious and terrible ones. One I like is How To Be An Adult In Relationships.