As a PhD student in ME I find this collections of statements wrong.
You need physical concepts in most engineering disciplines. For ME you will obviously be focusing on Mechanics (i.e. 1st semester physics - Newtonian Mechanics mostly). EE electricity and magnetism will be more of a focus. Then of course the conceptual understanding of matter and chemical interaction would be the most important for a ChE.
The other engineering disciplines (Aero/Astro, Civil, Industrial, etc.) are really offshoots of these basic three (maybe not Industrial, but that is another topic). Aero and Civil really have roots in ME so to be successful in those fields you should really focus on physics and particularly the mechanics side of physics.
IMHO, as an engineer, I feel that the most important thing a student can do is focus on conceptual understanding and that will provide the most success to you in the future. Focusing all of your energy on math will likely just make you good at math, but not necessarily help with engineering. In engineering math is a tool which allows us to describe/model physical interactions and phenomenon; furthermore it is a single tool in your arsenal. The most important tool you develop in college as an engineer is intuitions and insights, not math, so focus on the concepts (of course that does not mean you should not develop good math skills those are important too).
That being said, I would recommended studying hard for physics. There isn't any easy way it just takes lots of practice. With practice you will begin to develop intuition and conceptual understanding of the subject which will serve you greatly when you get to college. If you need help, talk with your teacher, post here, or do some online research.