For example, for word $w$ at position $pos \in [0, L-1]$ in the input sequence $\boldsymbol{w}=(w_0,\cdots, w_{L-1})$, with 4-dimensional embedding $e_{w}$, and $d_{model}=4$, the operation would be
$$\begin{align*}e_{w}' &= e_{w} + \left[sin\left(\frac{pos}{10000^{0}}\right), cos\left(\frac{pos}{10000^{0}}\right),sin\left(\frac{pos}{10000^{2/4}}\right),cos\left(\frac{pos}{10000^{2/4}}\right)\right]\\
&=e_{w} + \left[sin\left(pos\right), cos\left(pos\right),sin\left(\frac{pos}{100}\right),cos\left(\frac{pos}{100}\right)\right]\\
\end{align*}$$
where the formula for positional encoding is as follows
$$\text{PE}(pos,2i)=sin\left(\frac{pos}{10000^{2i/d_{model}}}\right),$$
$$\text{PE}(pos,2i+1)=cos\left(\frac{pos}{10000^{2i/d_{model}}}\right).$$
with $d_{model}=512$ (thus $i \in [0, 255]$) in the original paper.
This technique is used because there is no notion of word order (1st word, 2nd word, ..) in the proposed architecture. All words of input sequence are fed to the network with no special order or position (unlike common RNN or ConvNet architectures), thus, model has no idea how the words are ordered. Consequently, a position-dependent signal is added to each word-embedding to help the model incorporate the order of words. Based on experiments, this addition not only avoids destroying the embedding information but also adds the vital position information. In the case of RNNs, we feed the words sequentially to RNN, i.e. $n$-th word is fed at step $n$, which helps the model incorporate the order of words.
This article by Jay Alammar explains the paper with excellent visualizations. Unfortunately, its example for positional encoding is incorrect at the moment (it uses $sin$ for the first half of embedding dimensions and $cos$ for the second half, instead of using $sin$ for even indices and $cos$ for odd indices).
