iO from Matrix Branching Programs
The problem is that two partial evaluations of same length $\ell$ will have the form
$R_0 B_1 R_\ell^{-1}$
and
$R_0 B_2 R_\ell^{-1}$.
Thus, because the attacker possible knows $B_1$ and $B_2$, they can learn something
about the branching program that is obfuscated.
Encoding these randomized matrices with the multilinear map hides (or at least tries to hide) these relations.
Consider the security definition of iO.
It says that an attacker chooses two equivalent branching programs $P_1$ and $P_2$, sends them to an obfuscator, and then receives an obfuscated program $P$. The goal is to guess if $P$ corresponds to $P_1$ or $P_2$.
Now, if the obfuscation is done without the multilinear map, an attacker can break the security definition as follows:

Choose two different matrices $A$ and $B$;

Choose a length-2 branching program $P_1$ with a single input bit as follows:
$$\text{bit 0:} ~ ~ ~ ~ I  ~ ~ ~ ~ A$$
$$\text{bit 1:} ~ ~ ~ ~ I  ~ ~ ~ ~ B$$
$$\text{inp}(1) = \text{inp}(2) = 1$$

Choose a length-2 branching program $P_2$ with a single input bit as follows:
$$\text{bit 0:} ~ ~ ~ ~ A  ~ ~ ~ ~ I$$
$$\text{bit 1:} ~ ~ ~ ~ B  ~ ~ ~ ~ I$$
$$\text{inp}(1) = \text{inp}(2) = 1$$


Notice that $P_1(0) = P_2(0) = A$ and $P_1(1) = P_2(1) = B$, thus, they are equivalent programs.
Now, if $P$ obfuscates $P_1$, it is of the form
$$\text{bit 0:} ~ ~ ~ ~ R_1 \cdot R_2^{-1} ~ ~ ~ ~ ~ ~ ~ ~ R_2 \cdot A \cdot R_3^{-1}$$
$$\text{bit 1:} ~ ~ ~ ~ R_1 \cdot R_2^{-1}  ~ ~ ~ ~ ~ ~ ~ ~ R_2 \cdot B \cdot R_3^{-1}$$
Thus, we can distinguish the obfuscated programs by checking if the matrices
corresponding to the first step are equal.