It is the combination of two suffixes.
-lA- can be added to a noun or adjective to make a transitive verb. So:
yara (n., a wound) -- yaralamak (v., to injure someone)
-n- can be added to a transitive verb to make it intransitive/reflexive/passive (these distinctions of English grammar do not perfectly match the Turkish categories)
yaralamak (v., to injure someone) -- yaralanmak (v., to get/be injured)
That is the conceptual structure, and there are many triples that follow this form. For another example, açık (adj., clear) -- açıklamak (v. to clarify/explain) -- açıklanmak (v. to be explained) However, you aren't quite free to do it to any word you want. The language has already "decided" which of these transformations make recognizable words and which do not. So you can easily see how hasta (adj., sick/ill) becomes hastalanmak (v., to fall ill/get sick), but contrary to what you might assume, the word "hastalamak" does not exist. If you want to say "to make someone else sick" you would have to say "hasta etmek" or similar.
I don't know much about grammar but as a native speaker, I can talk about the words you were asking about:
Kırıllanmak sounds seriously wrong, as the initial -ıl after "kır" (break) already makes the expression passive. So "Kırılmak" can be translated as "to be/get broken" both in the literal and metaphorical senses (kalbim kırıldı -> my heart has been broken). If there had been a word kırıllanmak, the root of it would have been kırıl, not kır.
As for ilgilemek and acılanmak; they don't make much sense and are not used but don't sound awfully wrong. Especially acılanmak could somewhat mean to [passively] receive pain, perhaps by an unfortunate event like someone's loss. I mean those words, unlike kırıllanmak, don't look grammatically wrong but weird or otherwise having a bit of forced-to-create feeling. Perhaps you may come across acılanmak in a poem or song.
I looked up kullanmak for you. As I expected, the root is "kul" which means a servant / slave. Problem is that kullanmak means to use :( Frankly I can't see any semantic connection between "kul" and "kullanmak". My best bet is that the meaning of its root is completely lost, so "kullan" part has somewhat become the "new" root. This explains a lot, for example you can add -ıl to its root to make it "kullanılmak" which is a very frequently used word, again, both literally (to be used) and metaphorically (to be used / abused).
Definitely, treating the word as if the root is "kullan" fits quite well in how the verbs work generally in Turkish. Kullanmak usually takes a direct object, yes.
About generalizing -la+n, I am not sure if I can entirely help besides giving some additional examples. Note that invisibibleteacup's comment does an extremely well job in explaining how it works.
Ünlenmek: Becoming famous. Ünlemek doesn't make sense.
Belirmek: To appear. Belirlemek: To detect; to realize something as distinct. Belirlenmek: In passive form of the previous. "kurallar belirlendi" would mean "the rules are set/decided upon/coined.". "düşman belirlendi" would mean the enemy is detected or is now distinctly known to us etc.
Gizlemek: To hide. Gizlenmek: To become hidden.
Gözlemek: To observe. Gözlenmek: to be observed.
Yaşlanmak: Becoming old. Well, apparently no such thing as yaşlamak.
I'm still trying to generalize the meaning of -lan. :-) But that's where half the fun is, not that I mind some help.
You may be right about kullan taking on its own meaning, since it takes a direct object (right?) and -lan isn't supposed to (right?). But it began with "causing something to be your servant."
And thank you to both zubiz and invisibleteacup. I've copied the examples into my notes.
I can't think why you'd even see this sentence in early levels. Also, for what it's worth, I completed Babbel.com's Turkish grammar before I began Duolingo. D is great but it does not explain basics enough to actually learn them. I would have been wholly lost w/o a better foundation. Babbel stops at about halfway through Duolingo's program, though, so this does take me farther---and lets me ask questions like this.
Alright here it is: -la is a suffix to add a quality to the root: to give a new state; to make start doing/being.
Therefore -n being the passive AND the reflexive suffix, -lan roughly adds a sense of to be made start doing/being (either by someone else -passive, or by themselves -reflexive).
All the examples will fit this but while some roots don't allow -la, the other won't allow -lan, while others allow both cases. This is all to do with prevalence: Some forms just didn't stick or just don't make sense. You'll see.
To show some examples:
BAŞ (n. head/start) -LA (v. to start, to give it a start) -N (v. to be started)
DAR (adj. narrow, difficult) -LA (v. to disturb, to give one difficulty) -N (v. to be disturbed)
DIŞ (n. out) -LA (v. to cast out, to give one out) -N (to be cast out)
KURU (adj. dry) -LA (v. to dry, to give it a dry state) -N (v. to dry themselves, to be made dry)
GİZ (n. secrecy) -LE (v. to hide, to give a secrecy) -N (v. to make themselves secret)
SU (n. water) -LA (v. to give water) -N (v. to water up, to well up)
As for the non-standard additions (non-standard case between #'s)
HOR (onom. snorring sound) -LA (v. to snorr, to make snorring sound) ###-N (v. to be snorred)###
PAT (onom. explosion sound) -LA (v. to make exploding sound, to explode) ###-N (v. to be exploded)###
[Passive cases in these don't make sense in English either- not wrong per se, but meaningless]
YAŞ (n. age) ###-LA (v. to make old)### -N (v. to age, to be made old [by themselves], to get old)
[Here, aging is an automatic process, so it goes straight to the reflexive verb. NOW, if we're talking about making someone old on Photoshop or something, the word for it SHOULD HAVE BEEN yaşlamak, but people just decided to get on with yaşlandırmak (which is grammatically redundant but the prevalent word.]
KUL (n. servant) ###-LA (v. to give a servant, to make on served)### -N (v. to use)
[In this case, the -la addition is not standard, or prevalent.]
I also wrote: "I was sick" and it was marked wrong telling me that it must be "I got sick". As I am not a native English speaker and in German we mostly are saying "Ich war krank" and you would hear only in very seldom cases "Ich bin krank geworden or ich wurde krank", but never "Ich bekam krank" May be in English it is reverse as in German. But most of my mistakes in these lessons are due to my English. As I want to learn Turkish here and not English, it is not such important for me.