Video solution:

Click to load, then click again to play

Written solution:

Each number is added once per line it is on. Every point is on $2$ lines except for $B$ which is on $3.$

This means $$\begin{align*} 2A + 3B &+ 2C + 2D \\&+ 2E + 2F = 47, \end{align*}$$ so $$\begin{align*} &2(A + B + C + D + E + F) \\ &= 47-B. \end{align*}$$

Since $A,B,C,D,E,F$ are unique digits from $1$ to $6,$ each digit is represented exactly once, making $$\begin{align*} &A + B + C + D + E + F \\ &= 21. \end{align*}$$

With $$\begin{align*} &2(A + B + C + D + E + F) \\ &+ B = 47, \end{align*}$$ we know $B + 2(21)=47,$ so $B=5.$

Thus, the correct answer is E.