We have
$$\begin{aligned} x + y + xy &= 3 \\ \Leftrightarrow xy + x + y + 1 &= 4 \\ \Leftrightarrow (x + 1)(y + 1) &= 4. \end{aligned}$$
Therefore we can rewrite the inequality in the following way:
$$\begin{aligned} &x+y \ge 2 \\ \Leftrightarrow &x+1+y+1 \ge 4 \\ \Leftrightarrow &x+1+y+1 \ge 2\sqrt{(x+1)(y+1)}. \end{aligned}$$
The last line is an immediate consequence of the AM-GM inequality.
In the last step, equality holds exactly for $x+1=y+1$, i.e. $x=y$. Taking into account the equality $x+y=2$ we obtain $x=y=1$ which is indeed an admissible case of equality and thus the only one.