The aim of shrinkage of parameters is to obtain better generalisation for your linear estimator.
It is not true that you get lower variance with an OLS estimator. If you have a lot of data an OLS estimator is likely to perform better than a RIDGE or LASSO estimator.
We are interested in smaller variance on unseen data. If we start to model the noise with our linear model then on new samples, we will have larger error, because noise in this interpretation cannot be modelled.
I think the easiest way to interpret why shrinkage is useful is to consider that your data has some inherent noise. If this inherent noise has an impact larger than some of your coefficients, then it would be safe to assume that those coefficients just fit the noise in the data, and not the effects that can be modelled by your linear model.
Another way to understand is by the principle of Occam's razor. Say that your linear model is able to explain the same data point with multiple linear combinations of the functions that you've used in your linear estimator. If two solutions are equivalent, a sparser solution is desired, because it uses less of your functions to model the effects.
In some cases (i.e. when you have less data than parameters, so called ill-conditioned problem) you can't even do ordinary least squares, and in that LASSO and RIDGE still works.