The best tool I've discovered to help students make these connections is the written word. I hand out the facts about motion graphs, and we do my version of a graph-matching exercise. But students can't just get the answer right and call it a day... they must write their justifications using (a) a fact from the sheet written out word-for-word, and (b) how that fact connects to the graph their working on.
And in this way I can nip faulty reasoning in the bud. I make them rewrite immediately when they tell me "the cart slows down because the graph says," because there's no fact of physics involved there - even if they're right that the cart slows down.
More importantly, I pay careful attention to subjects and verbs. The graph can change steepness; the graph's vertical axis value can change. The steepness and vertical axis values represent how a cart in the classroom moves. It's important that no one says "the graph moves" or "the cart's steepness changes." When I see those statements, I ask the student to rewrite with the correct subject and verb.
Students at first find this nitpicky. So what. By now they should (and do) know that physics isn't about right answers, physics is about communicating an understanding of how the world works. After a few classes, the class is really quite good at interpreting motion graphs, and they stop confusing the features of the representation with the real, live motion of a cart.
The following is a note I sent to my 9th grade class last night as a reminder of the care they must use in their written responses to motion graph questions.
Please consider carefully the subjects and verbs you use on your motion graph justifications.
"The cart moves closer to zero on the vertical axis, and so slows down" makes no sense. As you've seen, the cart moves on a track in the classroom; the cart cannot move anywhere "on the vertical axis."
"The graph moves upward on the vertical axis" similarly makes no sense - the graph does not move, the graph is still on the paper on your desk.*
* Unless you threw it upward or something.
The cart moves; the graph does not. The vertical axis of the graph indicates how fast and which way the cart moves.
You want to say, "The vertical axis values get closer to zero, so the cart's speed gets closer to zero."
(And you never, ever want to use the word "it." Write "I didn't say it" on top of your problem set for an extra credit point. Don't tell others this, keep it to yourself! :-) )