So, you need to learn how to develop this insight. It's not easy, but maybe just knowing what you are looking for (that is, to develop intuitive insight into what you are learning) puts you into the right track, and hopefully it gets easier with practice later.
Here are a few resources. Please note that these are not resources for math, but resources on how to develop insight into math.
- Cal Newport's essay about insight in technical college courses: http://calnewport.com/blog/2008/11/14/how-to-ace-calculus-th... (the general idea still aplies even if you're not studying in/for college)
- BetterExplained Math Cheatsheet - http://betterexplained.com/cheatsheet/ (a collection of intuitive, insight-generating explanations on a variety of math topics, from basic to advanced)
- Coursera's Learning How To Learn course - https://www.coursera.org/course/learning
- A Mind for Numbers book - http://www.amazon.com/Mind-For-Numbers-Science-Flunked-ebook... (the book used in the course above)
I hope this helps. A few basic, general tips for developing insight is to think about applications. For example, your slope of line problem. If instead you called the y-axis "distance travelled" and the x-axis time, can you "feel" that the slope of the line is the speed? What if you substitute "distance travelled" for "revenue generated"? What would be the equivalent of "speed" in this case? Try with a few more examples, and hopefully you will develop an insight that the slope of the line is the rate of change of something, and will be able to apply it to many situations.