Let's assume that your signal is indeed composed of two parts: even and odd:
$$s(t)=s_e(t)+s_o(t)$$
We also know following properties of these functions:
- Even: $f(-x)=f(x)$
- Odd: $f(-x)=-f(x)$
Let's calculate the time inversion of your signal $s(-t)$ and apply above properties:
$$s(-t)=s_e(-t)+s_o(-t)=s_e(t)-s_o(t) $$
So now let's do the trick and add the: $s(t)$ and $s(-t)$:
$$s(t)+s(-t) = \color{blue}{s_e(t)}+s_o(t) + \color{blue}{s_e(t)}-s_o(t)=\color{blue}{2s_e(t)} $$
Solve it for $\color{blue}{s_e(t)}$, and you will get:
$$\color{blue}{s_e(t)}=\dfrac{s(t)+s(-t)}{2} $$
In the end let's subtract $s(t)$ and $s(-t)$:
$$s(t)-s(-t) = s_e(t) + \color{red}{s_o(t)} - s_e(t)+\color{red}{s_o(t)}=\color{red}{2s_o(t)} $$
Rearrange and you will get:
$$\color{red}{s_o(t)}=\dfrac{s(t)-s(-t)}{2} $$