Q:

What a Contradiction?

This contradiction, you must see,
please take a guess, what must I be?
To toss the outside, and keep the rest...
but keep the outside and toss the mess!
No strings attached, they are in your teeth,
What edible object must I be?

A:

You are

 corn

because

 you take off the husk of corn, eat the kernels, then throw away the cob.

A:

I can think of a few possibilities but... are you perhaps a  

Popsicle?
I remove the wrap, eat all the icy goodness then trash the messy sticky stick.  

