In a previous answer, it was stated that the producers said that they felt that British/Irish accents "suited" the show. I'm going to explain why they feel that way.
Out-of-universe background on in-universe history
Although this question is not a duplicate thereof, this answer to a related question pretty much sums it up: Although the books and the TV series are only related and are not equivalent (and they have been drifting further apart for quite some time), the author, George R. R. Martin, did take heavy inspiration from the English Wars of the Roses (see Rolling Stone interview) — Stark←York and Lannister←Lancaster is just too obvious to ignore: The author isn't stupid and so he obviously meant that as a direct "hat-tip" to English history. Furthermore, in the same interview, he explains that one of the main the "boundaries" which define Westerosi "civilization" (and therefore by extension Westerosi "history" — since history is generally written by people with enough money/power to win wars and write about them) is inspired directly by Hadrian's Wall (see The Journal article).
Now, although the TV series Game of Thrones is only based on GRRM's A Song of Ice and Fire and is not a direct television production thereof, this background story still stands in some form. What this means is that the show is firmly grounded in the genre of Medieval fantasy (i.e. old-timey England plus magic and dragons!).
Out-of-universe motivation: Sociolinguistics!
Now, just because the series is directly inspired by English/British history doesn't necessarily mean that the actors have to speak with modern British accents. However, using British accents in (English) Medieval fantasy is a well-established tradition: Fantasy of this type typically draws heavily on contemporary views of Medieval European life (even J. R. R. Tolkien himself, who is basically the father of mainstream fantasy, drew heavily from the history, mythology and linguistics of the British Isles) and framing these stories in a British accent gives us (especially people from the "New" World) a feeling of them being "old" — which fits because the author typically wants the reader to associate their world with Medieval times: If the actors all spoke in a "Wild West" accent from Texas in the 1800s, it would feel "weird" because it's an accent which is heavily associated with a completely different place and era. On the other hand, "Wild West"-style linguistic influences don't feel out of place in the Firefly universe at all because it is very obviously inspired by Westerns and lacks associations with "Old Europe" — which Medieval fantasy does have.
What this means is that modern British dialects/accents serve in fact as a "genrelect" associated with Medieval fantasy: They are a sort of sociolect defining a character's connection to a particular genre. Just as it's weird to hear an old person use slang associated with younger speakers, it would be weird to hear a character in a Medieval fantasy universe speak, like, totally like a Valley Girl.
Moreover, the connotations associated with different English accents function just as well internally to the given fantasy universe as they do in our universe: The features of Bronn's speech mark him as being from a class lower than Tyrion, despite that Peter Dinklage is actually American and is just putting on a completely arbitrary pseudo-"South-eastern English aristocrat" English accent — Regardless of how "aware" the actor is of British sociolinguistic idiosyncrasies, he does know how to sound generically "posh" to viewers. If the actors just randomly used any accent (and actually, they still are pretty random compared to actual geographic/social divisions in the UK), we would lose the useful background information we get by hearing how they speak. Finally, the characters themselves have been proven to have the same sociolinguistic knowledge we do as viewers: Namely, when Tywin Lannister corrects Arya's attempts at speaking like a commoner, he states that a commoner would say m'Lord rather than my Lord. This mirrors sociolinguistic contrasts in real-life England both today and in the past: Using reduced forms such as m'Lord and usage of the me-possessive me Lord is associated with a lower-class background, whereas — on the flip side — using the non-reduced form my Lord is exclusive to educated classes: Someone from the upper crust may talk about huntin' an' shootin' when they feel like it, but the non-educated would always speak that way — hence why it was a shibboleth which gave Arya away.
The series is based on fantasy. Most people are used to hearing fantasy characters speak like the British, so anything else would be weird. Therefore, in order to let us recognize the characters' individual backgrounds/social class, the actors try to affect different English dialects which roughly correspond to equivalent backgrounds/classes in modern Britain.