Here's what it means to like something ironically:
You love greek food. One might assume you must like olives. If you in fact do not like olives you might tell people that you hate olives...but only ironically, (since you love Greek food). You would say this so that others will not make untrue assumptions about you (i.e. that you don't like Greek food on the basis you don't like olives)
Or, you love McDonald's Big Macs, but otherwise dislike McDonald's food, their business model, or fast food in general. If someone found out you like bigmacs, you might tell them you like them ironically. It allows you to keep you broader views or likes/dislikes in tact, emphasizing this fondness for bigmacs is the exception rather than the rule. Like the olive example, you might say you like them ironically so that people don't make untrue assumptions about you (i.e. that you are not a healthy person, that you buy-into globalization, etc. etc.)
Because you like something ironically does not necessarily mean you are ashamed or embarrassed about it (olive example), but it can certainly happen that way.
When it comes to bands and music, one might say they like a band, or a particular song by a band, but only ironically because they generally do not like, say, the genre of the band as a whole, but like the band anyway. They might say they like that band ironically, to keep their views (likely negative) about that genre or band in tact or save face. You might also say you like a band ironically if you know the person you are talking to has strong negative views about the band or genre to highlight it is an exception.
Typically, you are more likely to say that you like something ironically if it is more than just the genre you dislike, but what the genre represents. For example, if you are against pre-fabricated bands or music produced solely because it what sells because you believe that it is not true art, but you like, say Nickleback, you might say you like them ironically to ensure people do no think you support that way of making music.
Because it is cool to like "ture art", and not so cool to like prefab formulaic music, one might say they like Nickleback ironically when infact they like a bunch of differnet bands or music that is made that way.
Hipsters, on the other hand, do not like or care about anything. If they like or care about anything at all, they will say they do so ironically, so as to ensure that one does not think that there is actually anything in this world that else this person likes or cares about. However, hipsters think it's cool to be ironic, meaning they care about being ironic, which in fact is ironic in itself. As a result, hipsters may say they like something that they actually don't like just to seem ironic. Ironic, huh?
Hope that helps.