I see three things that make communication hard for you
- you are an introvert
- your colleagues don't understand your work and couldn't follow when you tried to explain it
- you don't know where your colleagues might need your help
By a delightful coincidence, the exact same technique helps with all three of these issues: ask questions.
When you ask questions, as soon as you stop talking it is someone else's turn to talk. You can keep a conversation going with far less effort than it would take to just offer facts at each other in turns. Most introverts find conversations hard and think they have to be the fact-fact-fact kind. But actually, the "taking turns saying facts" kind of conversation is just one variant. It's most common when the two people have the same level of information and interest in a subject. Imagine two sports fans:
A: I can't believe they are not starting Johnson for this game. B: I know, it's ridiculous! They really need his arm out there. A: For sure, and that kid is no substitute. B: Yeah, really, what is he, 14? A baby!
They can talk this way because they both know who Johnson is, what his arm is like and why the team needs it, and who that kid is. If one of those sports fans was to try to have that conversation with my mother it would go more like this:
Mum: Are you excited about the game? A: Yeah, but I can't believe they are not starting Johnson. Mum: Starting Johnson? Is he a lawnmower or something? A: No, I mean he's not one of the starting players. The coach decided to use someone else. Mum: Well isn't that the coach's job? A: Yes, but I think the coach is wrong. The team really needs his arm. Mum: Come on, don't they all have arms? A: I mean his ability to throw the ball. Against the team we're playing today I think that's more important than any of the other skills.
And so on. My mum is learning about the sport in question, and A is doing most of the talking. You could use this technique to join lunch conversations. Then when you've learned enough about whatever they're discussing to see an application of your work, you could ask some more questions:
So is the issue that the calculation takes too long, or that the answer isn't accurate enough?
You could occasionally toss in a sentence that refers to the benefits of your work, such as "My techniques can let you run the calculations over a far larger sample, which will increase your accuracy." Because of the context you've established in the conversation, they'll understand the importance of your point in a way they might not in a lecture that starts with the history of the field, covers the basics of how to do something simple, and ends with a conclusion about how much the accuracy can be increased. When people don't understand details they can still understand conclusions, especially when expressed in the words and jargon they prefer to use. Learning this jargon comes from joining conversations and really listening, as well as asking questions when you don't understand.
And eventually you might go to your boss and say
I think there's something I can do to help the X group. They have a problem with Y where it can't Z. I think if I A, B, and C that should solve it. Do you think I should suggest it to them? Who should I talk to?
I think that's what your boss is hoping you will do. Along the way people will come to understand what your area of expertise can do for them, even if they never understand how it works. And you will learn more about the applications of your techniques to real world problems, including what those problems actually are and how they affect companies and people.