How to simplify this block diagram to get the given transfer function?
Let \$g_1 = {10 \over s+1}, g_2 = {1 \over s}\$, and let \$w\$ be the output of the \$g_1\$ block.

You get \$w = g_1 (u - 2 w -g_2 w)\$, which gives
\$w = {g_1 \over1 + 2 g_1 + g_1 g_2} u\$, and so
\$y = {g_2 g_1 \over1 + 2 g_1 + g_1 g_2} u\$, or
\$y = {1  \over 1+{2 \over g_2}+{1 \over g_1 g_2}} u\$ if you would prefer.

Substituting in the values for \$g_1, g_2\$ gives the transfer function you wanted.