Let's say you are interested in the causal effect of $D$ on $Y$. The following statement are not quite precise but I think convey the intuition behind the two approaches:
Back-door adjustment: Determine which other variables $X$ (age, gender) drive both $D$ (a drug) and $Y$ (health). Then, find units with the same values for $X$ (same age, same gender), but different values for $D$, and compute the difference in $Y$. If there is a difference in $Y$ between these units, it should be due to $D$, and not due to anything else.
The relevant causal graph looks like this:
Front-door adjustment: This means that you need to understand precisely the mechanism by which $D$ (let's now say it's smoking) affects $Y$ (lung cancer). Let's say it all flows through variable $M$ (tar in lungs): $D$ (smoking) affects $M$ (tar), and $M$ (tar) affects $Y$; there is no direct effect.
Then, to find the effect of $D$ on $Y$, compute the effect of smoking on tar, and then the effect of tar on cancer - possibly through backdoor adjustment - and multiply the effect of $D$ on $M$ with the effect of $M$ on $Y$.
The relevant causal graph looks like this (where $U$ is not observed):
Here, front-door adjustment works because there is no open back-door path from $D$ to $M$. The path $D \leftarrow U \rightarrow Y \leftarrow M$ is blocked. This is because the arrows "collide" in $Y$. So the $D \rightarrow M$ effect is identified.
Similarly, the $M \rightarrow Y$ effect is identified because the only back-door path from $M$ to $Y$ runs over $D$, so you can adjust for it using the back-door strategy.
In sum, you can identify the "submechanisms", and there is no direct effect, so you can piece together the submechanisms to estimate the overall effect.
This will not work if $U$ infuences $M$, because then identifying the submechanisms does not work.
