I don't think there's a much better way to explain it than Duolingo does. It's not complicated, but its meaning can be pretty variable. In this case, you might think that "manĝigis" means "caused the object to be eaten", but it actually means "caused the object to be fed". As far as I can tell this is specific to that verb and not a function of the way -ig/-iĝ works in general.
The suffix -ig- is used to describe an action that makes do, or makes be. For example, manĝi = to eat, manĝigi = to feed (make eat). Another example would be esti = to be, estigi = to create, make, bring into being. It can also be used to turn intransitive verbs into transitive verbs. For example, bruli = to burn (be burning) and bruligi = to burn (something). It can be a verb on it's own, igi.
The suffix -iĝ- is used to describe a becoming. For example, ruĝa = red, ruĝiĝi = to blush (become red). It can also be used to describe a change in state. For example, stari = to stand (be standing), stariĝi = to stand up (become standing). Lastly (as a suffix), it can be used to turn a transitive verb into an intransitive one. For example, rompi = to break (something), rompiĝi = to break (become broken). Similiarly to igi, it can also be a verb on it's own, iĝi.
I hope everything is clear now :)
Please don't consider this usage to be good Esperanto. As a lot of you have noticed, it is really confusing.
It's allowable, not strictly wrong, and parallels other languages (including English), but the normal way to say "feed one's children" is "manĝigis al siaj idoj."
More generally, the default meaning of taking a transitive verb (like fari) and adding igi (->farigi) is "igi farata." If you want to say "igi faranta," well, you could say that - la birdo igis siajn idojn manĝataj - or just use al.
Please just use al.
The details are in PAG 119(C).
I think looking at another transitive verb you’re familiar with from the course getting suffixed with -ig- may help.
Consider, from Vallienne’s Ĉu li? (1908):
“Kiam oni havas kunmanĝantojn, oni ne devas ilin atendigi.”
Translation: When one has dinner companions, one shouldn’t keep them waiting.
Atendi, “to wait, wait for”, is transitive. In the naïve understanding, that means it shouldn’t take -igi, since it’s “already transitive”, but “should” take -iĝi-, “to make it intransitive”. For instance, here’s the basic transitive form with direct object in accusative that we all know:
Akompanu min. Ni atendu la buson kune.
Translation: Join me. Let’s wait for the bus together.
However, the objectless transitive atendi already works for any purpose one might reasonably attach to -iĝi-; not only, the agentive objectless “wait”:
Akompanu min! La tutan matenon, mi atendis. Ĉu la buso neniam alvenos?
Translation: Join me! All morning, I’ve been waiting. Will the bus never arrive?
but also the non-agentive “[a]wait”, for which one could imagine needing the nonexistent *atendiĝi:
Venu. La buso atendas.
Translation: Come. The bus awaits.
Given this, since any possible meaning of atendiĝi seems hazy (a semi-reflexive “I await”, with no object? Something metaphorically like “to be in limbo” or “to be in a holding pattern”?), it’s not in use.
Atendigi, on the other hand, isn’t exactly a common construction, but it’s found 20 times in La Tekstaro’s corpus. And its meaning is much clearer: “to make/cause/allow one(s) to wait”—i.e., “to keep waiting”, “to hold up, delay, or ignore, such that the object must wait.”
Same here— “la birdo manĝigis [ilin]”, the bird “makes/causes/allows [them] to eat”—it feeds them.