Sounds like you relaxed into the form (doing it without trying so hard - causing tension). It is a natural step forward but it has traps...for example you still have to be true to the general form (don't get sloppy). Many worry about the exact placement of the body, but my teachers just say let the center drive and let the whole body move...basically it is not exact but as long as the principles are there and you know the application (body moves differently depend on what you are doing with the same move in tai chi) then you should be ok.
Making the form fit "your natural movement" has dangers if taken the wrong way...consider the natural movement like the movement of a small child - what adults do normally is not natural movement. But this is a further step and not really related with what you mentioned - just something you may want to ponder.
"Make the form yours" also have many meanings. Yes, relaxing into the form and letting the body move (as mention above) is a natural step. But there is other steps in making the form yours. I believe that the form should be kept pure and relatively unchanged as it is taught to beginners, but the form does look subtly different when you have several more advanced people doing the exact same form. Much of the differences come down to intention and application. My teachers ask me what am I doing with it, before they will tell me whether I am doing it correctly. And each movement in the form has many applications and uses, so eventually you will have your own set in which will be different than someone else's. My fishes in eight to single whip transition (in Yang Lu Chan form) is different than most, as my application and use is different than most. It is basically the same movement, however my body drives it differently for the application...also my application is smaller so when I teach the form I must remember to open up the movement and make it large (disregarding my application as it does not work large). There are many a subtle strike or movement some students make more obvious because it is part of their application. I think this is part of making the form yours. You may even have an application other students and your teacher doesn't know about, so then that part of the form is really only yours.
There is a lot to think about in what your teacher said, and I am sure there is more than what I mentioned. More will occur to you as your understanding grows...just keep it in the back of your head.
~ Eric Putkonen
(Teaching Tai Chi Chuan in Plymouth, Minnesota)