Applied to linear regression case.
Multicollinearity happens when your predictors are linearly dependent (or close to be). This means, some of your N predictors can be obtained (or nearly) by linear combinations of the others. If predictor A is linearly dependent, you can remove it, and the ability to fit of your system remains the same. If it is not exactly linearly dependent, your ability to fit data will be lower generally.
Related to "the predictive power of a model is not influenced by multicollinearity". You can fit your response variable Y and have the same error by using new orthogonal predictors as long as your collinear predictors define the same vector space as the new orthogonal predictors.
With an example. Imagine we are in 2 dimensional space.
We have two points: Y1 is (40,50), y2 is (39,50).
We want to approximate Y1 and Y2 (with error 0) using two bases.
Base E: e1, e2 are (1,0) and (0,1)
Base A: a1, a2 are (1,-1) and (1,-0.99)
e1 and e2 are orthogonal and are base vectors of the plane.
a1 and a2 are almost linear dependent, they have a strong collinearity, but they are also base vectors of the plane because they are not the same vector.
We observe the results of the coefficient estimates with Y1 and Y2 (small changes in data)
Y1 is fitted as 40*e1 + 50*e2 and Y2 is predicted as 39*e1 + 50*e2
Y1 is fitted as -8960*a1 + 9000*a2 and Y2 is predicted as -8861*a1 + 8900*a2
You can observe the size and variance of the coefficients when using predictors with strong collinearity. By the way, other choices of a1 and a2 may show more variance in this example.