Why do we pick random features in random forest
The idea of random forests is basically to build many decision trees (or other weak learners) that are decorrelated, so that their average is less prone to overfitting (reducing the variance). One way is subsampling of the training set. The reason why subsampling features can further decorrelate trees is, that if there are few dominating features, these features will be selected in many trees even for different subsamples, making the trees in the forest similar (correlated) again. 

The lower the number of sampled features, the higher the decorrelation effect. On the other hand, the bias of a random forest is the same as the bias of any of the sampled trees (see for example Elements of Statistical Learning), but the randomization of random forests restrict the model, so that the bias is usually higher than a fully-grown (unpruned) tree. You are correct in that you can expect a higher bias if you sample fewer features. So, "feature bagging" really gives you a classical trade-off in bias and variance.