Disclaimer: The question has been answered in related posts, example this one which addresses LSZ reduction formula without explaining the minus sign issue.
We consider a unitary operator $U(t+dt,t):\mathcal{H}\to\mathcal{H}$ which acts on states $|a;t\rangle$ to give $|a;t+dt\rangle$. From classical mechanics the Hamiltonian generates time-evolution, we propose that $U(t+dt,t)=1-\frac{i}{\hbar}Hdt$ which satisfies the desired properties 1). conservation of probability 2). transitivity of time evolution by composition.
Well but $1+\frac{i}{\hbar}Hdt$ does the job perfectly as well. Clearly as put in all textbooks $e^{\frac{i}{\hbar}Ht}|a;t=0\rangle$ will not evolve the state to $|a;t\rangle$ when $H$ is time-independent, it is $e^{-\frac{i}{\hbar}Ht}|a;0\rangle=|a;t\rangle$. But then how do we know which sign to pick in the first place?
