Multicollinearity increases the standard errors of the coefficients. Increased standard errors in turn means that coefficients for some independent variables may be found not to be significantly different from 0, whereas without multicollinearity and with lower standard errors, these same coefficients might have been found to be significant and the researcher may not have come to null findings in the first place. In other words, multicollinearity misleadingly inflates the standard errors. Thus, it makes some variables statistically insignificant while they should be otherwise significant.

It is like two or more people singing loudly at the same time. One cannot discern which is which. They offset each other.

Source: http://www.chsbs.cmich.edu/fattah/courses/empirical/multicollinearity.html