@ChrisGerig de Rham's book is certainly well written and I usually end up recommending it to everyone interested in currents. But that being said, it is not really a good introduction to what is now understood to be the theory of currents, as Federer and Fleming kind of totally transformed de Rham's ideas just a few years later.

One important note about Morgan's book is that at least its first half is meant as a kind of companion to Federer's magnum opus, i.e. they share the same notation and for many theorems Morgan will only give the important ideas and then point to the precise corresponding section of Federer's book, where all the technical details are carried out. Also, without wanting to spoil too much, the book is worth at least a look for the illustrations alone.

@HJRW Equally, one should also mention Gaspard Monge, who was on the same expedition to Egypt, similarly held political appointments during the Revolution and all the while also is known for his results in geometry and as being the father of optimal transport.

@SiddharthBhat It strongly depends on who you are talking to, that's why I deliberately chose to write it this way, because we tend to almost forget about it. I love differential forms, but I had to teach this stuff to electrical engineers for a while. Telling them about differential forms only gets you confusion and the eternal question "will it be on the exam?". But writing it out this way after having done all those "div curl = 0" and "find the potential" calculations for weeks will get you a lot of appreciation for the underlying structure.

@DavidWhite Case in point, I'm into PDE and while thanks to my own curiosity I know what a short exact sequence is, I never had to depend on that knowledge neither for research nor for any compulsory class and I'd guess the same holds for many people I know. That is not to say that we don't use them, in fact there is one that is used all the time, but naming a thing of which you only know a single example isn't really helpful.

I'd argue that this is one of the standard routes in applied analysis. Usually some theoretical physicist proposes equations for some physical phenomenon, someone from numerics does some preliminary simulations, displaying some interesting or strange behaviour, which then lures in the theorists who try to analyze and prove this behaviour of the equations. I'm not sure if this is in the spirit of the question though.

If this is not the main aim of your talk and you just want them to understand the idea, why not try "proof by experiment" and hand them a ball and a piece of string or a rubber band? It's not mathematical, but it will be more memorable than a proof which might be to advanced for their current level.

If you are mostly interested in the mathematical content, you might also try to translate them yourself. Statements and their proofs usually involve only a quite limited vocabulary, and the process is in a way self correcting, since you'll have enough mathematical knowledge to see if you end up with a correct proof. It will be a slow process in the beginning, but I'll guess a few papers in, you'll get quite fluent in reading mathematical German.