Is the language $0^n 1^n 0^n$ recursive?
Explaining intuitively why this language is recursive is easy: it's almost trivial to write a computer program in a high-level programming language (e.g. Java, C) that decides this language. Since TMs can simulate high-level languages, you get the result.

If you need a concrete TM that decides this language, try the following:

First, erase a 0 and go right until you encounter 1. Replace this 1 with some new symbol $X$.
Go back left, and repeat the process until you are out of 0s on the left.
Then go back right and do the same thing with the $X$s and 0s on the right.

If, at any point, something "unexpected" happens, (e.g. you run out of 1s, or have too many 1s), then reject. Otherwise you will end up with an empty tape, at which point you can accept.