The point of using activations is to enable the network to learn non-linear functions. If you do not use activations and just stack linear layers, the result is equivalent to a single linear layer. This is the mathematical proof with 2 linear layers (extendable to N by induction):
$ y = (xW_1 + b_1) W_2 + b_2 = x W_1 W_2 + (b_1 W_2 + b_2) = x W' + b'$
(where $W'= W_1 W_2$ and $b'= b_1 W_2 + b_2$)
Therefore, without activations, a multilayer neural network is equivalent to a linear regression model and therefore only available to learn linear functions over the feature space (lines in 2d, planes in 3D, etc).
While linear models are useful for some types of data, there are many other types of data where the relationship between input and output is non-linear.