Le is an Indirect Object. In Spanish, the Indirect Objects are me, te, le, nos, and os. For the verb gustar , you will always need an Indirect Object. You don't say "gusto" you say "me gusta". Because of the actual definition of the verb. Now, as for " le" and "se", they are the same thing, but you need to use " se" in front of another object pronoun that starts with L. Mira... "He gives it to them." "Él se lo/la (a ellos)." This was done in Spanish grammar to avoid a confusing double L sound. Any questions, comment here and I'll help if I can. You got this! :-)
The "a él" wouldn't represent "ing"... if present it would be there to signify that it's "he" that likes breaking things (as opposed to she or you). Without it you have to reply on context to know who is the vandal. Personally, in this situation I'd think that either "to break" or "breaking" are suitable translations of the meaning of the sentence.
It would be incorrect to say "He like." http://www.verb2verbe.com/conjugation/english-verb/like.aspx
In context, which is totally missing in DL's isolated sentences, you would probably know who is "liking" something. Then, you wouldn't need to add the "a él, a ella, etc." unless there might be some confusion about it. "A mi familia le gusta ir de vacaciones. A mi hermano le gusta ir a las montañas, pero me gusta ir a la playa." (My family likes to go on vacation. My brother likes to go to the mountains, but I like going to the beach.)
"Mi computadora portátil se rompió la semana pasada." My laptop broke last week. Try these for translations:
"Le" can mean "to him/her/it/you." (Remember that "gustar" sentences like this are backwards. Literally it would translate to: Breaking things is pleasing to someone.) Exactly who the "le" refers to could be cleared up by adding an introductory phrase such as "A él/ella/usted le gusta romper cosas." In this case, Duo chose not to do this, and since there is no context, you can take your pick: He/She/You/It(?) like(s) to break things.
Its my understanding that "romper" means to rip or tear whereas "quebrar" means to break. I know in Mexico quebrar is used exclusively for break and the 2 verbs are NOT interchangeable. Don't know if this is the same in Spain. I put "It likes to rip/tear things" and it was wrong.
"romper cosas" as a phrase is the subject of gustar here, and that is singular. The pluralness of "cosas" is embedded in that phrase, but doesn't extend outside of it. If you had multiple phrases, like "he likes to break things and fix them", it might be gustan, but I'm not 100% about that.