The vanishing gradient is best explained in the one-dimensional case. The multi-dimensional is more complicated but essentially analogous. You can review it in this excellent paper [1].
Assume we have a hidden state $h_t$ at time step $t$. If we make things simple and remove biases and inputs, we have
$$h_t = \sigma(w h_{t-1}).$$
Then you can show that
\begin{align}
\frac{\partial h_{t'}}{\partial h_t}
&= \prod_{k=1}^{t' - t} w \sigma'(w h_{t'-k})\\
&= \underbrace{w^{t' - t}}_{!!!}\prod_{k=1}^{t' - t} \sigma'(w h_{t'-k})
\end{align}
The factored marked with !!! is the crucial one. If the weight is not equal to 1, it will either decay to zero exponentially fast in $t'-t$, or grow exponentially fast.
In LSTMs, you have the cell state $s_t$. The derivative there is of the form
$$\frac{\partial s_{t'}}{\partial s_t} = \prod_{k=1}^{t' - t} \sigma(v_{t+k}).$$
Here $v_t$ is the input to the forget gate. As you can see, there is no exponentially fast decaying factor involved. Consequently, there is at least one path where the gradient does not vanish. For the complete derivation, see [2].
[1] Pascanu, Razvan, Tomas Mikolov, and Yoshua Bengio. "On the difficulty of training recurrent neural networks." ICML (3) 28 (2013): 1310-1318.
[2] Bayer, Justin Simon. Learning Sequence Representations. Diss. München, Technische Universität München, Diss., 2015, 2015.