Let $X$ be the number of pairs of shoes that are matched. It is somewhat complicated to work with $X$ directly, so we think of $X$ as a sum of simpler variables:
For $i=1,2,\ldots10$, let $X_i=\cases{1 & \text{if the }i\text{th pair is matched},\\ 0 & \text{otherwise}.}$
Then each $X_i$ is a Bernoulli variable and $X=\sum_{i=1}^{10}X_i$.
Expectation is linear, so $ \Bbb E(X)=\sum_{i=1}^{10}\,\Bbb E(X_i). $
Now we fix $i$ and find $\Bbb E(X_i) $:
Since $X_i$ is a Bernoulli variable, $\Bbb E(X_i)=P[X_i=1]$. But the probability that $X_i=1$ is the probability that the $i$th pair was matched. Since it is equally likely that any one of the other 19 shoes is paired with the left shoe of the $i$th pair, $P[X_i=1]={1\over19}$.
So $\Bbb E(X_i)={1\over19}$.
Finally, we have:
$\Bbb E(X)=\sum_{i=1}^{10}\Bbb E(X_i)=\sum_{i=1}^{10}{1\over19}=10/19.$