I'll admit that what I'm offering is an opinion, but as a former target audience member (biology graduate student who leaned more towards the physical science side -- and would sometimes attend math talks), this is my take on the situation and some suggestions that may help provide some inspiration for your own ideas.
Advice for Speaker Guidelines
Start by thinking of these specific talks as a seminar series, and discuss with your department what you want the theme of them to be. "Talk about applied math" is a broad theme that you can attempt to modify with guidelines, but I think 9 times out of 10, if a PI responds to your invitation to present, they'll just give their canned "this is my research" talk that all senior academics seem to have.
But if you instead send out a question/invitation that's something like this, you can at least start to have a PI think about how they may answer it. The key is that you give specific guidelines by way of asking them questions to address:
"CALL FOR SPEAKERS: "The most exciting thing about my research is ____"
[The part in quotation marks is the theme -- I'm using an open question, but something like "The #1 Puzzle in My Field" could also work.]
We are gathering speakers to talk to our first-year class of graduate students about the exciting opportunities in applied math, and we believe your work would conform well to our series. We are not looking for a research talk per se, but instead the reason why you're excited about the research you're conducting and the questions you're answering. Feel free to discuss what you do, but know that the purpose of these talks is to expose our graduate students to the big research questions in different fields and get some idea about how the tools and techniques of applied math can help tackle them.
[This sets the level and expectations of the talk by setting the "take-away" you want your audience to experience.]
"Your talk should be entirely self contained. You can assume the audience is mathematically mature and has a solid understanding of undergraduate real analysis and linear algebra but do not assume any prior knowledge in physics, biology or more specialized mathematics. Of course feel free to talk about such topics, just be sure to explain all basic concepts needed to follow the rest of the presentation as soon as they come up."
[Now that you've set your expectation, you can include these specific guidelines. This moves the audience from "interested lay public" to "budding professionals"]
[Conclude in some appropriate way; discuss compensation, accommodations, lab/group tours, etc. Discuss what you require of the speaker to properly advertise their talk -- see below]
The caveat, of course, is that this is more work and you may not get as many researchers willing to come in and speak (because ideally, the speaker would craft a talk specifically for this event -- or at least modify the aforementioned "this is my research" talk for it). Nevertheless, you'll likely self-select for researchers who are more keen towards public speaking and the kind of research outreach for which you're looking. Feel free to poll your students, too, about what they would like to learn from invited speakers.
For your graduate students
I can talk to this personally.
The standard trick that my department used was to offer bonus points in class for students who attended the department seminars (or, more painfully, required we attend and write a report on the offered talks). Obviously the former option is a better incentive -- if morale is an issue, forcing graduate students to do something at risk of penalty may only make things worse.
Therefore, I'd suggest looking at how you advertise the seminar talks. This is when having a specific theme for these talks is important. If students recognize that there's another talk in the "What excites me about my research is ____." series, then it at least gives them some expectation as to the quality -- and level -- of the experience.
I'd bet that 99% of the fliers departments put out to advertise talks include the title, a picture of the speaker, and then the abstract that the speaker submitted. Something that may be better is to ask the speaker to fill out an easily digested questionnaire that you can advertise instead:
"What's the outstanding question in my field? [ask them to answer in 15 words or less] "
"Why is this an important question? [30 words]
"What mathematical tools do I use to help answer this question? [just a list]"
To be honest, whenever I would read the abstracts posted for talks, those are the questions for which I'd be looking. If I saw "cryo-EM," then I knew I wouldn't enjoy the talk very much; if I saw "single-molecule dynamics simulations," then that would get me to pull out my calendar and mark the time and date.
Thank you for indulging a long answer, and thank you for being concerned about the morale of your students. If you have any other questions, feel free to let me know. Good luck!