I would like to understand why ($M$ and $N$ two manifolds, $x\in M,y\in N$ )
$$ T_{(x,y)}(M\times N) \cong T_xM\oplus T_yN$$
Actually, I don't have a very good understanding of what the direct sum of two vector spaces is.
And does it follow from the definition of a direct sum that the elements in the direct sum are orthogonal?
This direct sum you can see as a direct product, since if $W_1.W_2 \subseteq V$ are subspaces such that $W_1 \cap W_2 = \{0\}$, then $W_1 \times W_2 \cong W_1 \oplus W_2$. Becaus of this we sometimes write $\oplus$ even when thinking of the cartesian product.
That being said, consider $\Phi\colon T_{(x,y)}(M\times N) \to T_xM \oplus T_yN$ given by $$\Phi(v) = ({\rm d}(\pi_M)_{(x,y)}(v), {\rm d}(\pi_N)_{(x,y)}(v)).$$This is clearly linear. And it is also surjective: take $(v_1,v_2) \in T_xM\oplus T_yN$. Take curves $\alpha_1,\alpha_2$ such that $\alpha_1(0)=x$, $\alpha_2(0) = y$, $\alpha_i'(0) = v_i$. Define $\alpha(t) = (\alpha_1(t),\alpha_2(t))$. It is instructive to check that $\Phi(\alpha'(0)) = (v_1,v_2)$. Since domain and codomain have the same dimension, $\Phi$ is an isomorphism.