In Modern Quantum Mechanics by J. J. Sakurai, page 21, the identity operator is mentioned as
$1 =|+\rangle\langle+|+|-\rangle\langle -|$.
I know that $|+\rangle\langle+| =1$ but do not understand the $+$ in between the two outer product. Please explain.
The two orthonormal vectors which span this space are $|+\rangle$ and $|-\rangle$. A generic vector can be written $|\psi\rangle = a|+\rangle + b|-\rangle$, which one might also express as a column vector: $$|\psi\rangle = \pmatrix{a\\b}$$ The operator $|+\rangle\langle +|$ acts on the aforementioned vectors as $$\bigg(|+\rangle\langle +|\bigg) |+\rangle =|+\rangle\underbrace{\langle+|+\rangle}_{=1} = |+ \rangle \qquad \qquad \bigg(|+\rangle\langle +|\bigg)|-\rangle = |+\rangle\underbrace{\langle +|-\rangle}_{=0} = 0$$ As a result, we may write it in matrix form as $|+\rangle\langle +| = \pmatrix{1&0\\0&0}$. Similarly, $|-\rangle\langle -|=\pmatrix{0&0\\0&1}$. The sum of these two operators is $$|+\rangle\langle+| + |-\rangle\langle-| = \pmatrix{1&0\\0&0}+\pmatrix{0&0\\0&1} = \pmatrix{1&0\\0&1}=\mathbb I$$ where $\mathbb I$ is the identity operator.
