to understand what the inversion does, it's best to see that matrix as some kind of function that applies to points and vectors. That function takes the coordinates of that point/vector in one vector space, and returns the coordinates of that same point/vector in another vector space. The camera matrix typically tells how to transform points from screen space to world space. But that doesn't help you if you want to transform project points on your screen, because you need to transform points from world space to object space. So, you need the transformation in the opposite way that the one that the camera matrix gives you. That's why you need to invert the camera matrix to project a point.
If you invert a 3x3 matrix that only contains a rotation in 3D, it turns out that the inverted matrix is exactly the same as the transposed matrix. So, in that case you may use the cheaper transposition to get the inverted rotation (inversion is a more expensive operation).
However, we typically use 4x4 matrices for our 3D transformations because they can also represent translations of points. It turns out that if the transformation still only represents a rotation, then the inverted matrix equals to what you get if you only transpose the upper left 3x3 block of the 4x4 matrix (and thus leaving the right column and bottom line alone).
If the 4x4 matrix also represents a translation, it already gets a little trickier ... to get the inverted matrix, you can still transpose the upperleft 3x3 block, but the translation part (possible the 3 top elements in the right column, or the 3 left elements in the bottom row, depending what style of matrices you use) must also be taken care of. First see those 3 values as a 3D vector, rotate it by the transposed 3x3 block and negate it. Then put it back where it came from. Now you have you're inverted matrix.
In any other case (if matrix also represents scaling, skewing, ...), you should really invert the matrix instead of relying on transposing it.
I hope this is an answer to your questions.