$^\dagger$I personally find Susskind's explanation here unnecessarily confusing. Here is what I think he means to say. I proceed to give a qualitative description of what Susskind means (or so I think). If you want a more precise, mathematical exposition see below the line.
Recall that time evolution is governed by the Schrödinger equation. Equivalently, time evolution is enacted by a unitary operator. An important property of unitary operators in this context is that they have an inverse. This just means that time evolution is mathematically reversible--just act on your state with the inverse time evolution operator.
However, there is an important circumstance in Quantum Mechanics in which the time evolution is not unitary (i.e., not reversible). This is upon performing a measurement. We say that we "collapse" a wave function into a special state known as an eigenstate upon measuring with respect to some observable quantity. This "collapse" is not reversible, so it is not unitary.
What Susskind is then saying is that in the first scenario, we perform no measurement. We imagine the electron in some initial state and time evolve it forward unitarily. That means we can evolve it backwards in time and obtain exactly the initial state.
However, in the second scenario, we do perform a measurement with the screen that the electron hits. This measurement causes a "collapse" which is an irreversible evolution of our initial state. Hence, we can no longer reverse time (evolve time backwards) to get our original state.
Overall, the main point of Susskind here is that measurements on a quantum system deeply impact the system. This is contrary to measurements on classical systems (a motionless baseball stays a motionless baseball whether you look at it or not).
$^\dagger$ I tried to balance accuracy and level of detail. For example, the precise definition of a measurement is absent from my exposition, but I don't think it is necessary to include for the point OP would like to understand. Moreover, I should explicitly say that this exposition draws from assumptions made in textbook Quantum Mechanics. So, saying "collapse" is allowed.
I proceed to give a precise mathematical exposition of the situation Susskind outlines.
In Quantum Mechanics, a system is represented by a Hilbert space $\mathcal{H}$ as well as a collection of operators $\mathcal{L}(\mathcal{H})$. There is one distinguished operator $\hat{H} \in \mathcal{L}(\mathcal{H})$ known as the Hamiltonian. The Hamiltonian is the generator of time translations. This means that for $\lvert \psi(t=0) \rangle \in \mathcal{H}$ we have
$$\exp({-i\hat{H}t'})\lvert \psi(t=0) \rangle = \lvert \psi(t=t') \rangle$$
where $\exp({-i\hat{H}t'})$ is known as the time evolution operator. It is aptly named because it evolves a state forward in time. Because the time evolution operator is unitary, it has an inverse: $\exp({i\hat{H}t'})$, which evolves a state backwards in time.
All Susskind is saying is that in situation 1) we have
$$\exp({-i\hat{H}t'})\lvert \psi(t=0) \rangle \xrightarrow{\text{Reverse Time Evolution}} \exp({+i\hat{H}t'})\exp({-i\hat{H}t'}) \lvert \psi(t=0) \rangle = \lvert \psi(t=0) \rangle.$$
I.e. we unitarily evolve our initial state by time $t'$ and then unitarily evolve the resulting state backwards by time $t'$, resulting in exactly the original state.
In scenario 2), we have
$$ \exp({-i\hat{H}t'}) \lvert \psi(t=0) \rangle \xrightarrow{\text{Measurement}} \lvert E_n \rangle \xrightarrow{RTE} \exp({+i\hat{H}t'}) \lvert E_n \rangle \neq \lvert \psi(t=0) \rangle$$
unless the initial state was $\lvert E_n \rangle$ to begin with and $\lvert E_n \rangle$ is a simultanous eigenstate of both the Hamiltonian and observable you are measuring, i.e. unless you were measuring a conserved quantity and your initial state was an eigenstate of said conserved quantity.
