Hi Diseraph, good on you for reaching out for advice on how to support your trans friend. Just a quick point before we start - does your friend identify as male or female? This isn't about political correctness, it's actually about respecting your friend's identity. Being misgendered really makes you feel like crap, so getting things like that right can actually make a big difference toward feeling supported. Even asking your friend how they would like to be addressed shows that you care.
It's not surprising that your friend has not opened up a great deal. The sense of isolation and people not understanding your situation when coming out as trans, or transitioning, is really high. Just generally letting your friend know that you are available to talk anytime if they need support (if that is genuinely what you feel you can provide) is a good start, and they will open up to you when they're ready. Other than that, your husband's approach of treating your friend as a regular person, not a label, is also a good one.
It sounds like your main connection is over video games, so I would just keep on the way you have been doing before your friend disclosed to you about being trans. If you're keen to listen, then just listen, you'd be surprised how often that is all that is needed. Just a general open-ended 'how are things going', the same you would for any friend provides that opportunity. You don't have to try and solve your friend's problems. If you're feeling helpless, then you can imagine how your friend must be feeling. Just being able to listen helps to know that you're not alone, and that if they're having problems keeping friends in other areas of life, the fact that you're not running for the hills is important in reinforcing that not everyone is going to react badly to your friend's transition.
I hope some of that helps.