In order to understand how multiple decrement tables work, it is critical to understand the assumptions underlying the multiple decrement model that is used to produce the multiple decrement table.
A multiple decrement model is a multi state model with one active state and one or more absorbing states.
Independent rates:
Independent rates of transition assume that there is only one decrement acting on the population, i.e. there is only one way in which an individual may leave the active state.
For example, with the normal life tables that you have been using, death is the only decrement acting on the population. In other words, death is the only way in which people leave the active state of being alive.
Dependent rates:
Dependent rates take into account the competing forces of decrements acting on a population. So for example in pension schemes, a member may exist the active state of being an active member of the scheme via:

death,

withdrawal due to illness, changed jobs, etc,

retirement (early or on normal terms), etc.
So in this example, the rate (or probability) of death will depend on the withdrawal and retirement rates (or probabilities).
[Aside, one would expect that a person who existed from the active state due to illness will have a higher rate of mortality than another person of the same age who exited (retired) on normal conditions, hence the death rate is expected to be dependent on the other decrements.]
The equation above is derived by noting that under a multiple decrement model there is one active state and all other states are absorbing, this implies that the occupancy probability of the active state is equal to the probability of returning to the active state, i.e. \({}_tp_{x}^{\bar{a}\bar{a}} \) = \({}_tp_{x}^{aa} \), where \(a\) is the active state. So when considering the independent rates of transition from the active state to the absorbing states, we assume that all other decrements do not influence these rates, hence we can just apply the normal occupancy probability formula:
$$ p_{x}^{*j} = e^{\int_{0}^{1} \mu_{x+t}^{0j} dt} $$
which is obtained by solving the Kolmogorov (forward) differential equations.
Note that the implicit assumption above is that the independent rate of transition from the active state to state \(j\) is equal to the dependent rate of transition between the states.