We have
$$\frac{x+1}{y} + \frac{y+1}{x} = \frac{x+x+y}{y} + \frac{y+x+y}{x} = 2\left(\frac{x}{y} + \frac{y}{x}\right) + 2.$$
For $x, y > 0$, the AM-GM inequality gives
$$\frac{\frac{x}{y} + \frac{y}{x}}{2} \geq \sqrt{\frac{x}{y} \cdot \frac{y}{x}} = 1,$$
which immediately implies the desired inequality.

Equality holds for $\frac{x}{y} = \frac{y}{x}$, i.e. $x = y = \frac{1}{2}$.