What is the simplest scientific case that can be made for the fact that humans are changing the climate?
Understanding two indisputable facts and knowing some grade school-level vocabulary makes a very compelling case. That's just three simple things.
Simple Thing #1:
I'll start with the grade school vocabulary. You need to know the difference between weather and climate. Weather is about the state of the atmosphere with regards to temperature, moisture and wind at a given time in a given place. Climate is the weather over a long period of time; typically on a scale that is at least decades long. Knowing what the weather is tells you what clothes to wear today. Knowing what the climate is tells you what clothes you should own.
Oh, and you should understand the meaning of the word "global."
It may be cold in Washington this winter. That's the weather. And it's not global.
Simple Thing #2 (a.k.a., Simple Fact #1):
Carbon dioxide and other greenhouse gases absorb heat in the atmosphere and make it warm up. We can demonstrate that very, very easily in the laboratory.
Get yourself a seltzer bottle and try it at home. It really is that simple. I'd be more inclined to put the jars in a sunny spot than to use the lights, but it will work either way.
Of course, you'd be right to note that the concentration of carbon dioxide in the jar is considerably more than that of the atmosphere. Indeed. But, hello! The atmosphere is miles thick -- it's a heck of a lot bigger than that jar. And Earth is blanketed by all those miles of atmosphere that trap heat in.
Simple Thing #3 (a.k.a., Simple Fact #2):
Each and everyone of us (assuming that us means folks who are sitting in comfortable homes or offices while the chill of winter is outside our walls and that we've been in some sort of vehicle in the last few days) is adding ton after ton after ton of carbon dioxide to the atmosphere.
Really. Tons and tons. If you burned 10 gallons of gas in the last week or two, the stuff didn't just go away. You turned about 60 pounds of gasoline into about 180 pounds of carbon dioxide. Add to that the fuel to heat your house, to make and transport the things you own and have consumed, build your roads and other infrastructure, and to move your military about and have them do their (our?) business and pretty soon it adds up to lots and lots of carbon dioxide.
And there are hundreds of millions of us living this lifestyle and billions more who aspire to it.
(To refresh your memory on how burning stuff creates more weight of carbon dioxide than the weight of the stuff you started with, take a look back to this post with its nifty animations.)
There's more to it...
Of course you can get complicated pretty quickly, and to really understand climate change you need to read and understand more than the few paragraphs I've put together here. But the above really is beyond dispute. And understanding the above makes it very, very difficult to dispute that the stuff we humans are (and have been) up to is changing how the atmosphere works in ways that are making the planet heat up.
That's it: two little facts and a bit of grade school vocabulary. Is it simple enough for you?